Fractional Kolmogorov Equations with Singular Paracontrolled Terminal Conditions

TL;DR

This paper extends the theory of fractional Kolmogorov equations to handle singular terminal conditions and low-regularity drifts using paracontrolled methods.

Contribution

The novel contribution is a unified solution theory for singular and non-singular data using paracontrolled terminal conditions without requiring modified paraproducts.

Findings

A paracontrolled solution space is introduced that ensures regularity without modified paraproducts.

The approach generalizes previous results to singular paracontrolled terminal conditions.

The method applies broadly to linear PDEs using the paracontrolled ansatz.

Abstract

We consider backward fractional Kolmogorov equations with singular Besov drift of low regularity and singular terminal conditions. To treat drifts beyond the so-called Young regime, we assume an enhancement assumption on the drift and consider paracontrolled terminal conditions. Our work generalizes previous results on the equation from Cannizzaro and Chouk (Ann Probab 46:1710–1763, 2018), Kremp and Perkowski (Bernoulli 28:1757–1783, 2022. 10.3150/21-BEJ1394) to the case of singular paracontrolled terminal conditions and simultaneously treats singular and non-singular data in one concise solution theory. We introduce a paracontrolled solution space that implies parabolic time and space regularity on the solution without introducing the so-called modified paraproduct from Gubinelli and Perkowski (Commun Math Phys 349:165–269, 2017). The tools developed in this article apply for general linear PDEs that can be tackled with the paracontrolled ansatz.

Full text

Introduction

Kolmogorov equations are second-order parabolic differential equations. Their connection to stochastic processes was already investigated by Kolmogorov in the seminal work [4]. There exist analytic and probabilistic methods to study Kolmogorov equations. We refer to the books [5–8] for an overview on Kolmogorov equations in both finite- and infinite-dimensional spaces. In the finite-dimensional setting, Kolmogorov equations with bounded and measurable coefficients and uniformly elliptic diffusion coefficients can be treated as a special case of the infinite-dimensional Dirichlet form methods of [9], see also [5, Section 2.4.1] and the connection to the martingale problem in [5, Section 6.1.2]. We remain in the finite-dimensional setting, but consider distributional drifts in Besov spaces. Besov spaces play well with the paracontrolled calculus that defines products of distributions, cf. the Littlewood–Paley theory in [10]. Previous articles that consider distributional drifts are [11, 12], as well as [1] in the setting of rougher distributional drifts. Heat kernel estimates for the solution to the Kolmogorov equation were established in [13, 14].

In the article [2], the Laplace operator is replaced by a generalized fractional Laplacian. We extend our previous results on the equation from [2] to allow for irregular terminal conditions. That is, we consider the fractional parabolic Kolmogorov backward equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \big (\partial _{t}-\mathfrak {L}^{\alpha }_{\nu }+V\cdot \nabla \big )u=f,\quad u(T,\cdot )=u^{T}, \end{aligned}$$\end{document}

on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,T]\times \mathbb {R}^{d}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {L}^{\alpha }_{\nu }$$\end{document} generalizes the fractional Laplace operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^{\alpha /2}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2]$$\end{document} and V is a vector-valued Besov drift with negative regularity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (\frac{2-2\alpha }{3},0)$$\end{document} , i.e.  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\in C([0,T],(B^{\beta }_{\infty ,\infty })^{d})=C_{T}\mathscr {C}^{\beta }_{\mathbb {R}^{d}}$$\end{document} . Since V is a distribution, we need to be careful with well-definedness of the product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\cdot \nabla u$$\end{document} . The regularity obtained from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-(-\Delta )^{\alpha /2}$$\end{document} suggests that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(t,\cdot ) \in \mathscr {C}^{\alpha +\beta }$$\end{document} if right-hand side f and terminal condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}$$\end{document} are regular enough. Therefore we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla u(t,\cdot ) \in \mathscr {C}^{\alpha +\beta -1}$$\end{document} . Since the product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(t,\cdot ) \cdot \nabla u(t,\cdot )$$\end{document} is well-defined if and only if the sum of the regularities of the factors is strictly positive, we obtain the condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha +2\beta -1 >0$$\end{document} , equivalently \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta > (1-\alpha )/2$$\end{document} . We call this the Young regime, in analogy to the regularity requirements that are needed for the construction of the Young integral. However, we go beyond the Young regime, considering also the so-called rough regime \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (\frac{2-2\alpha }{3},\frac{1-\alpha }{2}]$$\end{document} . In the rough case, we employ paracontrolled distributions (cf. [15]) to solve the equation. The idea is to gain some regularity by treating u as a perturbation of the solution of the linearized equation with additive noise, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t w = \mathfrak {L}^{\alpha }_{\nu }w - V$$\end{document} . The techniques work as long as the nonlinearity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V \cdot \nabla u$$\end{document} is of lower order than the linear operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {L}^{\alpha }_{\nu }$$\end{document} , i.e. for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha > 1$$\end{document} or equivalently \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2-2\alpha )/3 < (1-\alpha )/2$$\end{document} . The price one has to pay to go beyond the Young regime is a stronger assumption on V. That is, we assume that certain resonant products involving V are a priori given. Those play the role of the iterated integrals in rough paths theory (cf. [16]). We then enhance V by that resonant product component and call the enhancement \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}$$\end{document} .

In [1, 2], only regular terminal conditions were considered, i.e.  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}\in \mathscr {C}^{\alpha +\beta }$$\end{document} in the Young regime and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}\in \mathscr {C}^{2(\alpha +\beta )-1}$$\end{document} in the rough regime. The right-hand side f can either be an element of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{T}L^{\infty }$$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=V^{i}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\dots ,d$$\end{document} . There are techniques available to treat less regular terminal conditions, cf. [3, Section 6]. With the help of those techniques, one can allow for terminal conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}\in \mathscr {C}^{(1-\gamma )\alpha +\beta }$$\end{document} in the Young case and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}\in \mathscr {C}^{(2-\gamma )\alpha +2\beta -1}$$\end{document} in the rough case for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [0,1)$$\end{document} , obtaining a solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{t}\in \mathscr {C}^{\alpha +\beta }$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t<T$$\end{document} and blow-up \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow T$$\end{document} . In this work, consider moreover singular paracontrolled right-hand sides f as well as singular paracontrolled terminal condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}$$\end{document} , which includes all cases mentioned above. Moreover, we can consider f and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}$$\end{document} more generally as elements of Besov spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {C}^{\theta }_{p}=B_{p,\infty }^{\theta }$$\end{document} with integrability parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document} . Examples for terminal conditions that we cover in the rough case include the Dirac measure, that is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}=\delta _{0}\in \mathscr {C}^{0}_{1}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}=V(T,\cdot )$$\end{document} . To be more precise, in the rough regime, we assume paracontrolled right-hand sides and terminal conditions,

$$[eqn]$$

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\prime }, f^{\prime }_{t}\in \mathscr {C}^{\alpha +\beta -1}_{p}$$\end{document} and remainders \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\sharp }_{t}\in \mathscr {C}^{\alpha +2\beta -1}_{p}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\sharp }\in \mathscr {C}^{(2-\gamma )\alpha +2\beta -1}_{p}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [0,1)$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\prime }_t$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\sharp }_t$$\end{document} we also allow a blow-up \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow T$$\end{document} . We prove existence and uniqueness of mild solutions of the Kolmogorov backward equation for singular paracontrolled data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(f,u^{T})$$\end{document} . The paracontrolled solution is an element of the solution space with blow-up \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} at terminal time T. Our main motivation for considering paracontrolled data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(f,u^{T})$$\end{document} came from defining additive functionals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{0}^{t}f(s,X_s)ds$$\end{document} of the Markov process X with generator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {G}^{V}=\partial _t-\mathfrak {L}^{\alpha }_{\nu }+V\cdot \nabla $$\end{document} , since

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{0}^{t}f(s,X_s)ds = u(t,X_t)-u(0,X_0)+M_t \end{aligned}$$\end{document}

for a martingale M and the solution u. Defining these additive functionals for certain paracontrolled f was crucial in developing a solution theory of weak solutions to the associated SDE in [17, section 4].

As a by-product of the main result, we prove a new commutator estimate for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\mathfrak {L}^{\alpha }_{\nu })$$\end{document} -semigroup, cf. Lemma 6, that allows to gain not only space regularity, but also time regularity. Thanks to Lemma 6 there is no need for the so-called modified paraproduct from [3, Section 6.1]. Moreover, we prove continuity of the Kolmogorov solution map and a uniform bound for the solutions considered on subintervals of [0, T] for bounded sets of data.

It is important to mention that the techniques we develop in this article are not limited to that particular equation and can be used to treat other linear singular PDEs with the paracontrolled approach. In this sense we see the Kolmogorov PDE as a model example.

The work is structured as follows. In Sect. 2 we introduce the generalized fractional Laplacian \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {L}^{\alpha }_{\nu }$$\end{document} and its semigroup. We prove semigroup and commutator estimates. In Sect. 3 we introduce the solution spaces and prove generalized Schauder and commutator estimates thereon. Finally, we solve the Kolmogorov equation with singular paracontrolled data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(f,u^{T})$$\end{document} in Sect. 4 and prove continuity of the solution map, as well as a uniform bound for the solutions on subintervals.

Preliminaries

Below we introduce some technical ingredients about Besov spaces and paraproducts that we will need in the sequel. We study estimates for the generalized fractional Laplacian and its semigroup, as well as, commutator estimates involving the paraproducts and the fractional semigroup.

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p_{j})_{j\geqslant -1}$$\end{document} be a smooth dyadic partition of unity, i.e. a family of functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{j}\in C^{\infty }_{c}(\mathbb {R}^{d})$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\geqslant -1$$\end{document} , such that

1. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{-1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{0}$$\end{document} are non-negative radial functions (they just depend on the absolute value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in \mathbb {R}^{d}$$\end{document} ), such that the support of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{-1}$$\end{document} is contained in a ball and the support of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{0}$$\end{document} is contained in an annulus;
2. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{j}(x):=p_{0}(2^{-j}x)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in \mathbb {R}^{d}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\geqslant 0$$\end{document} ;
3. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=-1}^{\infty }p_{j}(x)=1$$\end{document} for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in \mathbb {R}^{d}$$\end{document} ; and
4. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\operatorname {supp}(p_{i})\cap \operatorname {supp}(p_{j})=\emptyset $$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|i-j|>1$$\end{document} . We then define the Besov spaces for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p,q\in [1,\infty ]$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} B^{\theta }_{p,q}:=\{u\in \mathscr {S}':\Vert u\Vert _{B^{\theta }_{p,q}}=\big \Vert (2^{j\theta }\Vert \Delta _{j}u\Vert _{L^{p}})_{j\geqslant -1}\big \Vert _{\ell ^{q}}<\infty \}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{j}u=\mathscr {F}^{-1}(p_{j}\mathscr {F}u)$$\end{document} are the Littlewood–Paley blocks, and the Fourier transform is defined with the normalization \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varphi }(y):=\mathscr {F}\varphi (y):=\int _{\mathbb {R}^{d}}\varphi (x)e^{-2\pi i\langle x,y\rangle }dx$$\end{document} (and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {F}^{-1}\varphi (x)=\hat{\varphi }(-x)$$\end{document} ); moreover, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {S}$$\end{document} are the Schwartz functions and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {S}'$$\end{document} are the Schwartz distributions.

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{\infty }_{b}=C^{\infty }_{b}(\mathbb {R}^{d},\mathbb {R})$$\end{document} denote the space of bounded and smooth functions with bounded partial derivatives. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=\infty $$\end{document} , the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{\theta }_{p,\infty }$$\end{document} has the unpleasant property that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{\infty }_b \subset B^\theta _{p,\infty }$$\end{document} is not dense. Therefore, we rather work with the following space:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} B^{\theta }_{p,\infty }:=\{u\in \mathscr {S}'\mid \lim _{j\rightarrow \infty }2^{j\theta }\Vert \Delta _{j}u\Vert _{L^{p}}=0\}, \end{aligned}$$\end{document}

for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty _b$$\end{document} is a dense subset (cf. [10, Remark 2.75]). We also use the notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {C}^{\theta }_{\mathbb {R}^{d}}:=(\mathscr {C}^{\theta })^{d}=\mathscr {C}^{\theta }(\mathbb {R}^{d},\mathbb {R}^{d})$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {C}^{\theta -}:=\bigcap _{\gamma <\theta }\mathscr {C}^{\gamma }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {C}^{\theta +}=\bigcup _{\gamma >\theta }\mathscr {C}^{\gamma }$$\end{document} . Furthermore, we introduce the notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {C}^{\theta }_{p}:=B^{\theta }_{p,\infty }$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in \mathbb {R}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {C}^{\theta }:=\mathscr {C}^{\theta }_{\infty }$$\end{document} with norm denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \cdot \Vert _{\theta }:=\Vert \cdot \Vert _{\mathscr {C}^{\theta }}$$\end{document} .

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\leqslant p_{1}\leqslant p_{2}\leqslant \infty $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\leqslant q_{1}\leqslant q_{2}\leqslant \infty $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in \mathbb {R}$$\end{document} , the Besov space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{s}_{p_{1},q_{1}}$$\end{document} is continuously embedded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{p_{2},q_{2}}^{s-d(1/p_{1}-1/p_{2})}$$\end{document} (cf. [10, Proposition 2.71]). Furthermore, we will use that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in B^{s}_{p,q}$$\end{document} and a multi-index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathbb {N}^{d}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \partial ^{n}u\Vert _{B^{s-|n|}_{p,q}}\lesssim \Vert u\Vert _{B^{s}_{p,q}}$$\end{document} , which follows from the more general multiplier result from [10, Proposition 2.78].

We recall from Bony’s paraproduct theory (cf. [10, Section 2]) that in general for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \mathscr {C}^{\theta }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v\in \mathscr {C}^{\beta }$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta ,\beta \in \mathbb {R}$$\end{document} , the product $[eqn]$ , is well-defined in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {C}^{\min (\theta ,\beta ,\theta +\beta )}$$\end{document} if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta +\beta >0$$\end{document} . Denoting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{i}u=\sum _{j=-1}^{i-1}\Delta _{j}u$$\end{document} , the paraproducts are defined as follows

$$[eqn]$$

Here, we use the notation of [18, 19] for the para- and resonant products $[eqn]$ and $[eqn]$ .

In estimates we often use the notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\lesssim b$$\end{document} , which means that there exists a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} , such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\leqslant C b$$\end{document} . In the case that we want to stress the dependence of the constant C(d) in the estimate on a parameter d, we write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\lesssim _{d} b$$\end{document} .

The paraproducts satisfy the following estimates for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p,p_{1},p_{2}\in [1,\infty ]$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\leqslant 1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta ,\beta \in \mathbb {R}$$\end{document} (cf. [14, Theorem A.1] and [10, Theorem 2.82, Theorem 2.85])

$$[eqn]$$

So if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta + \beta > 0$$\end{document} , we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert u v\Vert _{\mathscr {C}^{\gamma }_{p}}\lesssim \Vert u\Vert _{\mathscr {C}^{\theta }_{p_{1}}}\Vert v\Vert _{\mathscr {C}^{\beta }_{p_{2}}}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma :=\min (\theta ,\beta ,\theta +\beta )$$\end{document} .

We define the generalized fractional Laplacian \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {L}^{\alpha }_{\nu }$$\end{document} via Fourier analysis as follows.

Definition 1

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,2)$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} be a symmetric (i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu (A)=\nu (-A)$$\end{document} ), finite and nonzero measure on the unit sphere \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\subset \mathbb {R}^{d}$$\end{document} . We define the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {L}^{\alpha }_{\nu }$$\end{document} as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathfrak {L}^{\alpha }_{\nu }\mathscr {F}^{-1}\varphi =\mathscr {F}^{-1}(\psi ^{\alpha }_{\nu } \varphi )\qquad \text {for }\varphi \in C^\infty _b, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi ^{\alpha }_{\nu } (z):=\int _{S}|\langle z,\xi \rangle |^{\alpha }\nu (d\xi ).$$\end{document} For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =2$$\end{document} , we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {L}^{\alpha }_{\nu }:=-\frac{1}{2}\Delta $$\end{document} .

Remark 1

If we take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} as a suitable multiple of the Lebesgue measure on the sphere, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi ^\alpha _\nu (z) = |2\pi z|^\alpha $$\end{document} and thus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {L}^{\alpha }_{\nu }$$\end{document} is the fractional Laplace operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^{\alpha /2}$$\end{document} .

Assumption 1

Throughout the paper, we assume that the measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} from Definition 1 has d-dimensional support, in the sense that the linear span of its support is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document} .

So far we defined \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {L}^{\alpha }_{\nu }$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty _b$$\end{document} , so in particular on Schwartz functions. But the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {L}^{\alpha }_{\nu }$$\end{document} on Schwartz distributions by duality is problematic, because for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,2)$$\end{document} the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi ^{\alpha }_{\nu }$$\end{document} has a singularity in 0. This motivates the next proposition.

Proposition 1

(Continuity of the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {L}^{\alpha }_{\nu }$$\end{document} ) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,2]$$\end{document} . Then for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in \mathbb {R}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u \in C^\infty _b$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \mathfrak {L}^{\alpha }_{\nu }u\Vert _{\mathscr {C}^{\beta -\alpha }_{p}}\lesssim \Vert u\Vert _{\mathscr {C}^{\beta }_{p}}. \end{aligned}$$\end{document}

In particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {L}^{\alpha }_{\nu }$$\end{document} can be uniquely extended to a continuous operator from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {C}^{\beta }_{p}$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {C}^{\beta -\alpha }_{p}$$\end{document} .

Proof

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \geqslant 0$$\end{document} it follows from [10, Lemma 2.2] the estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \mathfrak {L}^{\alpha }_{\nu }\Delta _j u\Vert _{L^p} \lesssim 2^{-j(\beta -\alpha )} \Vert u \Vert _{\mathscr {C}^{\beta }_{p}}$$\end{document} . This uses that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi ^{\alpha }_{\nu }$$\end{document} is infinitely continuously differentiable in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{d}\setminus \{0\}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\partial ^{\mu }\psi ^{\alpha }_{\nu }(z)|\lesssim |z|^{\alpha -|\mu |}$$\end{document} for a multi-index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathbb {N}_{0}^{d}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\mu |:=\mu _{1}+\dots +\mu _{d}\leqslant \alpha $$\end{document} and that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{j}u$$\end{document} has a Fourier transform, which is supported in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{j}{\mathscr {A}}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {A}}$$\end{document} is the annulus, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{0}$$\end{document} is supported. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=-1$$\end{document} we use that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\mathfrak {L}^{\alpha }_{\nu }\varphi = A\varphi $$\end{document} for test functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in C^{\infty }_{b}$$\end{document} and A defined as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A\varphi (x)=\int _{\mathbb {R}^{d}}\big (\varphi (x+y)-\varphi (x)-\textbf{1}_{\{|y|\leqslant 1\}}(y) \nabla \varphi (x) \cdot y\big )\mu (dy)\qquad \text {for }\varphi \in C_{b}^{\infty }. \end{aligned}$$\end{document}

and therefore

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&-\mathfrak {L}^{\alpha }_{\nu }\mathscr {F}^{-1}\tilde{p}_{-1}(x) \nonumber \\&\quad = \int _{\mathbb {R}^d}\left( \mathscr {F}^{-1}\tilde{p}_{-1}(x+y) - \mathscr {F}^{-1}\tilde{p}_{-1} u(x) - \nabla \mathscr {F}^{-1}\tilde{p}_{-1} u(x)\cdot y \mathbb {1}_{\{|y| \leqslant 1\}} \right) \mu (dy)\nonumber \\&\quad \lesssim \int _{B(0,1)} \Vert D^2\mathscr {F}^{-1}\tilde{p}_{-1}\Vert _{L^\infty }|y|^2 \mu (dy) + \Vert \mathscr {F}^{-1}\tilde{p}_{-1}\Vert _{L^\infty } \mu (B(0,1)^c) \lesssim 1 \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(0,1) = \{|y| \leqslant 1\}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{p}_{-1}$$\end{document} is smooth and compactly supported in a ball and such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{p}_{-1}p_{-1}=p_{-1}$$\end{document} . Then we obtain with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\mathfrak {L}^{\alpha }_{\nu }\Delta _{-1}u=-\mathfrak {L}^{\alpha }_{\nu }\mathscr {F}^{-1}\tilde{p}_{-1}*\Delta _{-1}u$$\end{document} and Young’s convolution inequality,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert -\mathfrak {L}^{\alpha }_{\nu }\Delta _{-1}u\Vert _{L^{p}}&\leqslant \Vert -\mathfrak {L}^{\alpha }_{\nu }\mathscr {F}^{-1}\tilde{p}_{-1}\Vert _{L^{1}}\Vert \Delta _{-1}u\Vert _{L^{p}}\\&\leqslant \Vert -\mathfrak {L}^{\alpha }_{\nu }\mathscr {F}^{-1}\tilde{p}_{-1}\Vert _{L^{\infty }}\Vert \Delta _{-1}u\Vert _{L^{p}}\lesssim \Vert u\Vert _{\mathscr {C}^{\beta }_{p}}. \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z \in \mathbb {R}^d \setminus \{0\}$$\end{document} , we also have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \psi ^{\alpha }_{\nu }(z)= |z|^\alpha \int _{S} \Big |\Big \langle \frac{z}{|z|},\xi \Big \rangle \Big |^\alpha \nu (d\xi ) \geqslant |z|^\alpha \min _{|y|=1} \int _{S} |\langle y,\xi \rangle |^\alpha \nu (d\xi ), \end{aligned}$$\end{document}

and by Assumption 1 the minimum on the right-hand side is strictly positive. Otherwise, there exists some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_0\ne 0$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _S |\langle y_0,\xi \rangle |^\alpha \nu (d\xi ) = 0$$\end{document} and this would mean that the support of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} (and thus also its span) is contained in the orthogonal complement of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\operatorname {span}(y_0)$$\end{document} .

Notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi ^{\alpha }$$\end{document} is none-smooth when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2)$$\end{document} and thus in general \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{-\psi ^{\alpha }}$$\end{document} is not a Schwartz function, even when excluding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=0$$\end{document} . In this case the proof of the semigroup estimates from [15, Lemma A.5/A.7] is actually not applicable. We give an alternative proof below.

Lemma 1

(Semigroup estimates) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} be a finite, symmetric measure on the sphere \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\subset \mathbb {R}^{d}$$\end{document} satisfying Assumption 1. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{t}\varphi :=\mathscr {F}^{-1}(e^{-t\psi ^{\alpha }_{\nu }}\hat{\varphi }) = \rho _t *\varphi $$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t > 0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _t = \mathscr {F}^{-1} e^{-t\psi ^\alpha _\nu } \in L^1$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in C^\infty _b$$\end{document} . Then we have for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta \geqslant 0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in \mathbb {R}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert P_{t}\varphi \Vert _{\mathscr {C}^{\beta +\vartheta }_{p}}\lesssim (t^{-\vartheta /\alpha } \vee 1) \Vert \varphi \Vert _{\mathscr {C}^{\beta }_{p}}, \end{aligned}$$\end{document}

and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta \in [0,\alpha ]$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert (P_{t}-\operatorname {Id})\varphi \Vert _{\mathscr {C}^{\beta -\vartheta }_{p}}\lesssim t^{\vartheta /\alpha }\Vert \varphi \Vert _{\mathscr {C}^{\beta }_{p}}. \end{aligned}$$\end{document}

Furthermore, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (0,\alpha )$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=\infty $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert (P_{t}-\operatorname {Id})\varphi \Vert _{L^{\infty }}\lesssim t^{\beta /\alpha }\Vert \varphi \Vert _{\mathscr {C}^{\beta }}. \end{aligned}$$\end{document}

Therefore, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta \geqslant 0$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{t}$$\end{document} has a unique extension to a bounded linear operator in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\mathscr {C}^{\beta },\mathscr {C}^{\beta +\vartheta })$$\end{document} and this extension satisfies the same bounds.

Proof

By [20, Lemma 11] and (7), we have that the stable density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _t=\rho (t,\cdot )$$\end{document} satisfies for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l=1,2$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |D^{l}_{y}\rho (t,y)|\le C t^{-l/\alpha } q_{\alpha }(t,y), \quad |\partial ^{l}_{t}\rho (t,y)|\le C t^{-l}q_{\alpha }(t,y), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q_{\alpha }(t,\cdot ))_{t>0}$$\end{document} is a family of probability densities that satisfy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{\alpha }(t,y)=t^{d/\alpha }q_{\alpha }(1,t^{-1/\alpha }y)$$\end{document} and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [0,\alpha )$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{\mathbb {R}^d}q_{\alpha }(t,y)|y|^{\gamma } dy\le C_{\gamma } t^{-\gamma /\alpha }$$\end{document} . In particular, it follows that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert D^{l}_{y}\rho (t,\cdot )\Vert _{L^{1}}\lesssim t^{-l/\alpha } \int _{\mathbb {R}^d}q(t,y)dy= t^{-l/\alpha }. \end{aligned}$$\end{document}

Moreover, we have by [10, Lemma 2.1] and Young’s inequality that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l\in \mathbb {N}_{0}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert P_{t}\varphi \Vert _{\mathscr {C}^{\beta +l}_{p}}\lesssim \Vert \Delta _{-1}P_{t}\varphi \Vert _{L^{p}}+\Vert D^{l}_{y}P_{t}\varphi \Vert _{\mathscr {C}^{\beta }_p}\lesssim (\Vert \rho _{t}\Vert _{L^{1}}+\Vert D^{l}_{y}\rho _{t}\Vert _{L^{1}})\Vert \varphi \Vert _{\mathscr {C}^{\beta }_p}. \end{aligned}$$\end{document}

Thus (8) follows in the case of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =l\in \mathbb {N}_{0}$$\end{document} . The general case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta \ge 0$$\end{document} follows by interpolating between the bound for l and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l+1$$\end{document} , where l is such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l\le \vartheta \le l+1$$\end{document} :

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Delta _{j}P_{t}\varphi \Vert _{L^p}&\lesssim \Vert \varphi \Vert _{\mathscr {C}^{\beta }_p}\min (2^{-j(\beta +l+1)}(t^{-(l+1)/\alpha }\vee 1),2^{-j(\beta +l)}(t^{-l/\alpha }\vee 1))\\&\lesssim 2^{-j(\beta +\vartheta )}(t^{-\vartheta /\alpha }\vee 1)\Vert \varphi \Vert _{\mathscr {C}^{\beta }_p}, \end{aligned}$$\end{document}

using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min (a,b)\le a^{\lambda }b^{1-\lambda }$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in [0,1], a,b\ge 0$$\end{document} and choosing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda =\vartheta -l$$\end{document} .

To see (9), we use Proposition 1 and (8) to estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert (P_{t}-\operatorname {Id})\varphi \Vert _{\mathscr {C}^{\beta -\vartheta }_{p}}&=\bigg \Vert \int _{0}^{t}(-\mathfrak {L}^{\alpha }_{\nu })P_{r}\varphi dr\bigg \Vert _{\mathscr {C}^{\beta -\vartheta }_{p}}\\ &\leqslant \int _{0}^{t}\Vert (-\mathfrak {L}^{\alpha }_{\nu })P_{r}\varphi \Vert _{\mathscr {C}^{\beta -\vartheta }_{p}}dr\\ &\lesssim \int _{0}^{t}\Vert P_{r}\varphi \Vert _{\mathscr {C}^{\beta -\vartheta +\alpha }_{p}}dr \\ &\lesssim \Vert \varphi \Vert _{\mathscr {C}^{\beta }_{p}}\int _{0}^{t} (r^{(\vartheta -\alpha )/\alpha }\vee 1) dr \lesssim \Vert \varphi \Vert _{\mathscr {C}^{\beta }_{p}} t^{\vartheta /\alpha }, \end{aligned}$$\end{document}

using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta \in (0,\alpha ]$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =0$$\end{document} , (9) trivially follows. (10) follows from (9) applied for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =\beta + \varepsilon \in (0,\alpha )$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =\beta -\varepsilon \in (0,\alpha )$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} is small, and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert (P_{t}-\operatorname {Id})\varphi \Vert _{L^{\infty }}&\le \sum _{j}\Vert \Delta _j (P_{t}-\operatorname {Id})\varphi \Vert _{L^{\infty }} \\&\lesssim \Vert \varphi \Vert _{\mathscr {C}^{\beta }}\bigg [\sum _{j:\, 2^{-j}< t} 2^{-j\varepsilon } t^{(\beta +\varepsilon )/\alpha } + \sum _{j:\, 2^{-j}\ge t} 2^{j\varepsilon } t^{(\beta -\varepsilon )/\alpha }\bigg ] \\&\lesssim \Vert \varphi \Vert _{\mathscr {C}^{\beta }} t^{\beta /\alpha }. \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The next three lemmas deal with commutators between the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\mathfrak {L}^{\alpha }_{\nu })$$\end{document} operator, its semigroup and the paraproduct. The proofs can be found in Appendix A.

Lemma 2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \mathscr {C}^{\sigma }_{p}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\in \mathscr {C}^{\varsigma }$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varsigma \in \mathbb {R}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document} . Then the commutator for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\mathfrak {L}^{\alpha }_{\nu })$$\end{document} follows:

$$[eqn]$$

Lemma 3

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_{t})$$\end{document} be as in Lemma 1. Then, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in (0,1)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varsigma \in \mathbb {R}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta \geqslant -\alpha $$\end{document} the following commutator estimate holds true, where the bound is uniform for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in (0,1]$$\end{document} :

$$[eqn]$$

Lemma 4

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {L}^{\alpha }_{\nu }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_{t})_{t\geqslant 0}$$\end{document} be defined as in Definition 1 and Lemma 1 and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2]$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T>0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in (0,1)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varsigma \in \mathbb {R}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \geqslant 0$$\end{document} . Then the commutator on the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\mathfrak {L}^{\alpha }_{\nu })P_{t}$$\end{document} follows:

$$[eqn]$$

The mild formulation of the Kolmogorov equation is given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_{t}=P_{T-t}u_{T}+\int _{t}^{T}P_{r-t}(V_{r}\cdot \nabla u_{r}-f_{r})dr=:P_{T-t}u_{T}+J^{T}(V\cdot \nabla u-f)(t). \end{aligned}$$\end{document}

Due to the Schauder estimates, considering a singular terminal condition with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{T}\in \mathscr {C}^{\beta +}_{p}$$\end{document} , we obtain that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert P_{T-t}u_{T}\Vert _{\mathscr {C}^{\alpha +\beta }_{p}}$$\end{document} blows up for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow T$$\end{document} and the blow-up is of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} . This motivates the definition of blow-up spaces below, from which we can build the solution space in the next section.

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T>0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}\in (0,T]$$\end{document} , and a Banach space X, let us define the blow-up space

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathscr {M}_{\overline{T},T}^{\gamma }X:=\{u:[T-\overline{T},T)\rightarrow X\mid t\mapsto (T-t)^{\gamma }u_{t}\in C([T-\overline{T},T),X)\}, \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert u\Vert _{\mathscr {M}_{\overline{T},T}^{\gamma }X}:=\sup _{t\in [T-\overline{T},T)}(T-t)^{\gamma }\Vert u_t\Vert _{X}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {M}_{\overline{T},T}^{0}X:=C([T-\overline{T},T),X)$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}=T$$\end{document} , we use the notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {M}^{\gamma }_{T}X:=\mathscr {M}_{T,T}^{\gamma }X$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta \in (0,1]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} , we furthermore define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C_{\overline{T},T}^{\gamma ,\vartheta }X:=\biggl \{u:[T-\overline{T},T)\rightarrow X\biggm | \Vert f\Vert _{C_{T}^{\gamma ,\vartheta }X}:=\sup _{0\leqslant s<t< T}\frac{(T-t)^{\gamma }\Vert f_{t}-f_{s}\Vert _{X}}{|t-s|^{\vartheta }}<\infty \biggr \} \end{aligned}$$\end{document}

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{T}^{\gamma ,\vartheta }X:=C_{T,T}^{\gamma ,\vartheta }X$$\end{document} . Let us also define for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta \in (0,1]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}\in (0,T]$$\end{document} , the space of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta $$\end{document} -Hölder continuous functions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-\overline{T},T]$$\end{document} with values in X,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C_{\overline{T},T}^{\vartheta }X:=\biggl \{u:[T-\overline{T},T]\rightarrow X\biggm | \Vert u\Vert _{C_{T}^{\vartheta }X}:=\sup _{T-\overline{T}\leqslant s<t\leqslant T}\frac{\Vert u_{t}-u_{s}\Vert _{X}}{|t-s|^{\vartheta }}<\infty \biggr \} \end{aligned}$$\end{document}

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{T}^{\vartheta }X:=C_{T,T}^{\vartheta }X$$\end{document} . We set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\overline{T},T}^{0,\vartheta }X:=C^{\vartheta }([T-\overline{T},T),X)$$\end{document} .

We have the trivial estimates

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u\Vert _{\mathscr {M}_{\overline{T},T}^{\gamma _{1}}X}\leqslant \overline{T}^{\gamma _{1}-\gamma _{2}}\Vert u\Vert _{\mathscr {M}_{\overline{T},T}^{\gamma _{2}}X},\quad \Vert u\Vert _{C_{\overline{T},T}^{\gamma _{1},\vartheta _{1}}X}\leqslant \overline{T}^{(\gamma _{1}-\gamma _{2})+(\vartheta _{2}-\vartheta _{1})}\Vert u\Vert _{ C_{\overline{T},T}^{\gamma _{2},\vartheta _{2}}X} \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\leqslant \gamma _{2}\leqslant \gamma _{1}<1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\vartheta _{1}\leqslant \vartheta _{2}\leqslant 1$$\end{document} . Moreover, we have that for a subinterval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-2\overline{T},T-\overline{T}]\subset [0,T]$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\overline{T}\leqslant \frac{T}{2}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u\Vert _{\mathscr {M}_{\overline{T},T-\overline{T}}^{0}X}\leqslant \overline{T}^{-\gamma }\Vert u\Vert _{\mathscr {M}_{T}^{\gamma }X}. \end{aligned}$$\end{document}

Schauder Theory and Commutator Estimates for Blow-Up Spaces

In this section, we define the solution space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{T,p}^{\gamma ,\alpha +\beta }$$\end{document} and prove Schauder and commutator estimates. We conclude the section with interpolation estimates for the solution spaces.

Heuristically, the solution space shall combine maximal space regularity (i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha +\beta $$\end{document} ) in a time-blow-up space with maximal time regularity (i.e. Lipschitz) in a space of low space regularity. By interpolation, the solution will then also admit all time and space regularities “in between".

Let us thus define for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in \mathbb {R}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document} , the space

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathscr {L}_{T,p}^{\gamma ,\theta }:=\mathscr {M}_{T}^{\gamma }\mathscr {C}^{\theta }_{p}\cap C_{T}^{1-\gamma }\mathscr {C}^{\theta -\alpha }_{p}\cap C_{T}^{\gamma ,1}\mathscr {C}^{\theta -\alpha }_{p}. \end{aligned}$$\end{document}

We moreover define for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathscr {L}_{T,p}^{0,\theta }:= C_{T}^{1}\mathscr {C}^{\theta -\alpha }_{p}\cap C_{T}\mathscr {C}^{\theta }_{p}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1}_{T}X$$\end{document} denotes the space of 1-Hölder or Lipschitz functions with values in X.

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}\in (0,T)$$\end{document} , we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{\overline{T},T,p}^{\gamma ,\theta }:=\mathscr {M}_{\overline{T},T}^{\gamma }\mathscr {C}^{\theta }_{p}\cap C_{\overline{T},T}^{1-\gamma }\mathscr {C}^{\theta -\alpha }_{p}\cap C_{\overline{T},T}^{\gamma ,1}\mathscr {C}^{\theta -\alpha }_{p}$$\end{document} and similarly \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{\overline{T},T,p}^{0,\theta }$$\end{document} .

The spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{T,p}^{\gamma ,\theta }$$\end{document} are Banach spaces equipped with the norm

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u\Vert _{\mathscr {L}_{T,p}^{\gamma ,\theta }}&:=\Vert u\Vert _{\mathscr {M}_{T}^{\gamma }\mathscr {C}^{\theta }_{p}}+\Vert u\Vert _{C_{T}^{1-\gamma }\mathscr {C}^{\theta -\alpha }_{p}} +\Vert u\Vert _{C_{T}^{\gamma ,1}\mathscr {C}^{\theta -\alpha }_{p}}\\&={\displaystyle \sup }_{t\in [0,T)}(T-t)^{\gamma }\Vert u_{t}\Vert _{\mathscr {C}^{\theta }_{p}}+\sup _{0\leqslant s<t\leqslant T}\frac{\Vert u_{t}-u_{s}\Vert _{\mathscr {C}^{\theta -\alpha }_{p}}}{|t-s|^{1-\gamma }} \\ &\quad +\sup _{0\leqslant s<t< T}\frac{(T-t)^{\gamma }\Vert u_{t}-u_{s}\Vert _{\mathscr {C}^{\theta -\alpha }_{p}}}{|t-s|}. \end{aligned}$$\end{document}

Notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \mathscr {L}_{T,p}^{\gamma ,\theta }$$\end{document} in particular implies that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\mapsto \Vert u_{t}\Vert _{\mathscr {C}^{\theta -\alpha }}$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-\gamma )$$\end{document} -Hölder continuous at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=T$$\end{document} .

The next corollary proves estimates for the semigroup \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_{t})$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\mathfrak {L}^{\alpha }_{\nu })$$\end{document} acting on the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{T,p}^{\gamma ,\theta }$$\end{document} . We will need the following auxiliary lemma. In particular, the lemma can be applied, to show that the inverse fractional Laplacian improves space regularity by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} (and not only by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta <\alpha $$\end{document} ). It is a slight generalization of [15, Lemma A.9, (A.1)]. Its proof can be found in Appendix A.

Lemma 5

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in \mathbb {R}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [0,1)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varsigma \geqslant 0$$\end{document} . Let moreover \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:\mathring{\Delta }_{T}\rightarrow \mathscr {S}'$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathring{\Delta }_{T}:=\{(t,r)\in [0,T]^{2}\mid t<r\}$$\end{document} , be such that there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} such that for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\geqslant -1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\leqslant t< r\leqslant T$$\end{document} , for the Littlewood–Paley blocks holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Delta _{j}f_{t,r}\Vert _{L^{p}}\leqslant C(T-r)^{-\gamma }\min (2^{-j\sigma },2^{-j(\sigma +\varsigma +\varepsilon \varsigma )}(r-t)^{-(1+\varepsilon )}). \end{aligned}$$\end{document}

Then it follows that for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in [0,T]$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \bigg \Vert \int _{t}^{T}f_{t,r}dr\bigg \Vert _{\mathscr {C}^{\sigma +\varsigma }_{p}}\leqslant [2C \max (\varepsilon ^{-1},(1-\gamma )^{-1})](T-t)^{-\gamma }. \end{aligned}$$\end{document}

Corollary 1

(Schauder estimates) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_{t})$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} be as in Lemma 1. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T>0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}\in (0,T]$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in [T-\overline{T},T]$$\end{document} we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^{T}v (t)=J^{T}(v) (t):=\int _{t}^{T}P_{r-t}v(r)dr$$\end{document} . Then we have for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in \mathbb {R}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta \in [0,\alpha ]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [\vartheta /\alpha ,1]$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert P_{T-\cdot }w\Vert _{\mathscr {L}_{\overline{T},T,p}^{\gamma ,\beta +\vartheta }}\lesssim \overline{T}^{(\gamma \alpha -\vartheta )/\alpha } \Vert w\Vert _{\mathscr {C}^{\beta }_{p}} \end{aligned}$$\end{document}

and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\leqslant \gamma '\leqslant \gamma < 1$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert J^{T}v\Vert _{\mathscr {L}_{\overline{T},T,p}^{\gamma ,\beta +\alpha }}\lesssim \overline{T}^{\gamma -\gamma '} \Vert v\Vert _{\mathscr {M}_{\overline{T},T}^{\gamma '}\mathscr {C}^{\beta }_{p}}. \end{aligned}$$\end{document}

Proof

For (18) we only prove the estimate in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\overline{T},T}^{1-\gamma }\mathscr {C}_{p}^{\beta +\vartheta -\alpha }$$\end{document} and in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\overline{T},T}^{\gamma ,1}\mathscr {C}_{p}^{\beta +\vartheta -\alpha }$$\end{document} , the estimate in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {M}^{\gamma }_{\overline{T},T}\mathscr {C}_{p}^{\beta +\vartheta }$$\end{document} follows from a direct application of Lemma 1.

Therefore we write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{T-t}w-P_{T-s}w=P_{T-t}(\operatorname {Id}-P_{t-s})w$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T-\overline{T}\leqslant s<t\leqslant T$$\end{document} and use Lemma 1 to conclude

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert P_{T-t}w-P_{T-s}w\Vert _{\mathscr {C}_{p}^{\beta +\vartheta -\alpha }}&\lesssim \Vert (\operatorname {Id}-P_{t-s})w\Vert _{\mathscr {C}_{p}^{\beta +\vartheta -\alpha }}\\ &\lesssim (t-s)^{1-\vartheta /\alpha }\Vert w\Vert _{\mathscr {C}_{p}^{\beta }}\\ &\lesssim \overline{T}^{(\gamma \alpha -\vartheta )/\alpha }(t-s)^{1-\gamma }\Vert w\Vert _{\mathscr {C}_{p}^{\beta }} \end{aligned}$$\end{document}

using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\leqslant \vartheta \leqslant \alpha $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \geqslant \vartheta /\alpha $$\end{document} . This controls \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert P_{T-\cdot }w\Vert _{C_{\overline{T},T}^{1-\gamma }\mathscr {C}_{p}^{\beta +\gamma -\alpha }}$$\end{document} . To bound the norm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert P_{T-\cdot }w\Vert _{C_{\overline{T},T}^{\gamma , 1}\mathscr {C}_{p}^{\beta +\gamma -\alpha }}$$\end{document} , we note that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert P_{T-t}w-P_{T-s}w\Vert _{\mathscr {C}_{p}^{\beta +\vartheta -\alpha }}&\lesssim (T-t)^{-\vartheta /\alpha }\Vert (\operatorname {Id}-P_{t-s})w\Vert _{\mathscr {C}_{p}^{\beta -\alpha }}\\ &\lesssim (T-t)^{-\vartheta /\alpha }(t-s)\Vert w\Vert _{\mathscr {C}_{p}^{\beta }}\\ &\lesssim \overline{T}^{(\gamma \alpha -\vartheta )/\alpha }(T-t)^{-\gamma }(t-s)\Vert w\Vert _{\mathscr {C}_{p}^{\beta }}. \end{aligned}$$\end{document}

To estimate the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {M}^{\gamma }_{\overline{T},T}\mathscr {C}_{p}^{\beta +\alpha }$$\end{document} -norm in (19), we use Lemma 5 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{t,r}=P_{r-t}v_{r}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma =\beta $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varsigma =\alpha $$\end{document} , to obtain for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in [T-\overline{T},T]$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (T-t)^{\gamma }\Vert J^{T}v(t)\Vert _{\mathscr {C}_{p}^{\beta +\alpha }}&=(T-t)^{\gamma -\gamma '}(T-t)^{\gamma '}\bigg \Vert \int _{t}^{T}P_{r-t}v_{r}dr\bigg \Vert _{\mathscr {C}_{p}^{\beta +\alpha }} \\&\lesssim \overline{T}^{\gamma -\gamma '}\Vert v\Vert _{\mathscr {M}_{\overline{T},T}^{\gamma '}\mathscr {C}_{p}^{\beta }}. \end{aligned}$$\end{document}

To prove the bounds on the time regularity in (19) we write

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} J^{T}(v)_{t}-J^{T}(v)_{s}=\int _{s}^{t}P_{r-s}v_{r}dr-(P_{t-s}-\operatorname {Id})\bigg (\int _{t}^{T}P_{r-t}v_{r}dr\bigg ), \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T-\overline{T}\leqslant s< t\leqslant T$$\end{document} . We can estimate by Lemma 1

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \bigg \Vert \int _{s}^{t}P_{r-s}v_{r}dr\bigg \Vert _{\mathscr {C}^{\beta }_{p}}&\leqslant \int _{s}^{t}\Vert P_{r-s}v_{r}\Vert _{\mathscr {C}^{\beta }_{p}}dr\\&\lesssim \Vert v\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\beta }}\int _{s}^{t}|T-r|^{-\gamma '}dr\\&\lesssim \Vert v\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\beta }}(|T-s|^{1-\gamma '}-|T-t|^{1-\gamma '})\\&\leqslant \overline{T}^{\gamma -\gamma '}|t-s|^{1-\gamma }\Vert v\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\beta }}, \end{aligned}$$\end{document}

using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\leqslant \gamma '\leqslant \gamma <1$$\end{document} and the estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |T-t|^{1-\gamma '}-|T-s|^{1-\gamma '}\leqslant |t-s|^{1-\gamma '}\leqslant \overline{T}^{\gamma -\gamma '}|t-s|^{1-\gamma }. \end{aligned}$$\end{document}

On the other hand, we can also estimate that term by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \bigg \Vert \int _{s}^{t}\!P_{r-s}v_{r}dr\bigg \Vert _{\mathscr {C}^{\beta }_{p}}&\lesssim \Vert v\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\beta }}\int _{s}^{t}\!|T-r|^{-\gamma '}dr\\ &\leqslant \Vert v\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\beta }}|T-t|^{-\gamma '}\int _{s}^{t}\!dr \\&\leqslant \overline{T}^{\gamma -\gamma '}\Vert v\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\beta }}|T-t|^{-\gamma }|t-s|. \end{aligned}$$\end{document}

Moreover, by Lemma 1 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =\alpha $$\end{document} and Lemma 5, we obtain that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \bigg \Vert (P_{t-s}-\operatorname {Id})\bigg (\int _{t}^{T}P_{r-t}v_{r}dr\bigg )\bigg \Vert _{\mathscr {C}^{\beta }_{p}}&\lesssim |t-s|\bigg \Vert \int _{t}^{T}P_{r-t}v_{r}dr\bigg \Vert _{\mathscr {C}^{\beta +\alpha }_{p}}\\ &\lesssim |t-s|\Vert v\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\beta }}(T-t)^{-\gamma '}\\ &\lesssim |t-s|\overline{T}^{\gamma -\gamma '}\Vert v\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\beta }}(T-t)^{-\gamma }, \end{aligned}$$\end{document}

and on the other hand we can estimate by Lemma 1 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =(1-\gamma )\alpha $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\bigg \Vert (P_{t-s}-\operatorname {Id})\bigg (\int _{t}^{T}P_{r-t}v_{r}dr\bigg )\bigg \Vert _{\mathscr {C}^{\beta }_{p}}\\&\quad \lesssim |t-s|^{(\alpha -\gamma \alpha )/\alpha }\bigg \Vert \int _{t}^{T}P_{r-t}v_{r}dr\bigg \Vert _{\mathscr {C}^{\beta +\alpha -\gamma \alpha }_{p}}\\ &\quad \lesssim |t-s|^{(\alpha -\gamma \alpha )/\alpha }\Vert v\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\beta }}\int _{t}^{T}(T-r)^{-\gamma '}(t-r)^{(\gamma \alpha -\alpha )/\alpha }dr\\ &\quad \lesssim |t-s|^{1-\gamma }\Vert v\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\beta }}(T-t)^{\gamma -\gamma '}\\ &\quad \lesssim |t-s|^{1-\gamma }\Vert v\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\beta }}\overline{T}^{\gamma -\gamma '}, \end{aligned}$$\end{document}

where we used that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma >0$$\end{document} and that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma '\leqslant \gamma <1$$\end{document} (if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0$$\end{document} , we can use the previous estimate instead). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Remark 2

A less general approach for dealing with singular initial conditions in paracontrolled equations was developed in [3]. The function spaces above (15) seem more flexible, and actually there is a mistake in the singular Schauder estimates in [3, Lemma 6.6]: Equation (49) therein is only true for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (0,2-\alpha )$$\end{document} , i.e. only for distributional initial conditions,1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (-\alpha ,0)$$\end{document} would force \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0=0$$\end{document} .

Next, we prove a commutator estimate for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^{T}$$\end{document} -operator and the paraproduct.

Lemma 6

(Commutator estimates) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T>0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}\in (0,T]$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varsigma \in \mathbb {R}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2]$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [0,1)$$\end{document} . Then for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \mathscr {C}^{\sigma }_{p}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v\in \mathscr {C}^{\varsigma }$$\end{document} the following semigroup commutator estimate holds

$$[eqn]$$

Furthermore, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\in \mathscr {L}_{\overline{T},T,p}^{\gamma ',\sigma }$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\leqslant \gamma '\leqslant \gamma <1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in C_T\mathscr {C}^{\varsigma }$$\end{document} , we have

$$[eqn]$$

Remark 3

It was already known that the commutator for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^{T}$$\end{document} -operator from the lemma allows for more space regularity than both of its summands. The above commutator estimate moreover yields a gain in time regularity, i.e. $[eqn]$ , provided that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\in \mathscr {L}^{\gamma ,\sigma }_{T}$$\end{document} .

Proof

Recall that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}^{\gamma ,\sigma +\varsigma +\gamma \alpha }_{T}$$\end{document} is equipped with the sum of the norms in

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathscr {M}_{T}^{\gamma }\mathscr {C}_{p}^{\sigma +\varsigma +\alpha \gamma },\quad C_{T}^{\gamma ,1}\mathscr {C}_{p}^{\sigma +\varsigma +\alpha \gamma -\alpha }\quad \text { and }\quad C_{T}^{1-\gamma }\mathscr {C}_{p}^{\sigma +\varsigma +\alpha \gamma -\alpha }, \end{aligned}$$\end{document}

that we need to estimate below.

For (20), the estimate in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {M}_{T}^{\gamma }\mathscr {C}_{p}^{\sigma +\varsigma +\alpha \gamma }$$\end{document} follows directly by the semigroup commutator Lemma 3 applied to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =\gamma \alpha $$\end{document} . For the estimate in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{T}^{\gamma ,1}\mathscr {C}_{p}^{\sigma +\varsigma +\alpha \gamma -\alpha }\cap C_{T}^{1-\gamma }\mathscr {C}_{p}^{\sigma +\varsigma +\alpha \gamma -\alpha }$$\end{document} we write for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\leqslant s\leqslant t\leqslant T$$\end{document} ,

$$[eqn]$$

The first summand we can estimate by the semigroup estimates (Lemma 1) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\operatorname {Id}-P_{t-s}$$\end{document} and the commutator estimate in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {M}_{T}^{\gamma }\mathscr {C}_{p}^{\sigma +\varsigma +\alpha \gamma }$$\end{document} , obtaining

$$[eqn]$$

This gives the estimate in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{T}^{\gamma ,1}\mathscr {C}_{p}^{\sigma +\varsigma +\alpha (\gamma -1)}$$\end{document} . Analogously we estimate the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{T}^{1-\gamma } \mathscr {C}_{p}^{\sigma +\varsigma +\alpha (\gamma -1)}$$\end{document} -norm using the Schauder estimates for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\operatorname {Id}-P_{t-s}$$\end{document} (obtaining a factor of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|t-s|^{1-\gamma }$$\end{document} ) and the commutator in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\overline{T},T}\mathscr {C}_{p}^{\sigma +\varsigma }$$\end{document} , i.e.

$$[eqn]$$

The second summand can be estimated using the semigroup commutator (Lemma 3) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =(\gamma -1)\alpha \geqslant -\alpha $$\end{document} and the semigroup estimate (9), such that

$$[eqn]$$

Using instead the semigroup commutator for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =-\alpha \geqslant -\alpha $$\end{document} and again the semigroup estimate (9) yields

$$[eqn]$$

Together, we obtain (20). For (21), we first prove that $[eqn]$ . To that end, we write

$$[eqn]$$

To estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{1}$$\end{document} , we utilize Lemma 5 for $[eqn]$ , where the assumptions of the lemma are satisfied by the semigroup commutator estimate (Lemma 3). Then, we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert I_{1}(t)\Vert _{\mathscr {C}^{\sigma +\varsigma +\alpha }_{p}}&\lesssim \Vert g\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\sigma }}\Vert h\Vert _{C_{T}\mathscr {C}^{\varsigma }}(T-t)^{-\gamma '}\\&\lesssim \overline{T}^{\gamma -\gamma '}\Vert g\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\sigma }}\Vert h\Vert _{C_{T}\mathscr {C}^{\varsigma }}(T-t)^{-\gamma }. \end{aligned}$$\end{document}

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{2}$$\end{document} , we apply Lemma 5 for $[eqn]$ . We check the assumptions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{t,r}$$\end{document} of that lemma, using the time regularity of g, as well as the paraproduct estimate (using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma -\alpha <0$$\end{document} ) and the semigroup estimates. Then, choosing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =0$$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =(1+\varepsilon )\alpha $$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \in [0,1]$$\end{document} , we estimate (the estimate is in fact valid for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \geqslant -\alpha $$\end{document} )

$$[eqn]$$

Applying Lemma 5 yields then the estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{2}$$\end{document} :

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert I_{2}(t)\Vert _{\mathscr {C}^{\sigma +\varsigma +\alpha }_{p}} \lesssim \overline{T}^{\gamma -\gamma '}\Vert g\Vert _{C_{\overline{T},T}^{\gamma ',1}\mathscr {C}^{\sigma -\alpha }_{p}}\Vert h\Vert _{C_{T}\mathscr {C}^{\varsigma }_{\mathbb {R}^{d}}}(T-t)^{-\gamma }. \end{aligned}$$\end{document}

Next, we prove the time regularity estimates on the commutator C(g, h). For that, we write for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T-\overline{T}\leqslant s\leqslant t\leqslant T$$\end{document} ,

$$[eqn]$$

where we define

$$[eqn]$$

and

$$[eqn]$$

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{s,t}:=g_{t}-g_{s}$$\end{document} and

$$[eqn]$$

We will consider the terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{st},B_{st}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{st}$$\end{document} separately and estimate each term in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\overline{T},T}^{1-\gamma }\mathscr {C}^{\sigma +\varsigma }$$\end{document} -norm and in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\overline{T},T}^{\gamma ,1}\mathscr {C}^{\sigma +\varsigma }$$\end{document} -norm.

We start with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{st}$$\end{document} , using the time regularity of g, obtaining on the one hand

$$[eqn]$$

using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma -\alpha <0$$\end{document} and Lemma 5 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{t,r}=P_{r-t}h_{r}$$\end{document} to bound the time integral. On the other hand, along the same lines, using instead \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\in C_{\overline{T},T}^{\gamma ',1}\mathscr {C}^{\sigma -\alpha }_p$$\end{document} , we can estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{st}$$\end{document} by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert B_{st}\Vert _{\mathscr {C}^{\sigma +\varsigma }_p}\lesssim \overline{T}^{\gamma -\gamma '}\Vert h\Vert _{C_{T}\mathscr {C}^{\varsigma }}\Vert g\Vert _{C_{\overline{T},T}^{\gamma ',1}\mathscr {C}^{\sigma -\alpha }_p}|t-s|(T-t)^{-\gamma }. \end{aligned}$$\end{document}

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{st}$$\end{document} , we use the semigroup commutator (Lemma 3) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =0$$\end{document} , as well as the time regularity of g, which yields

$$[eqn]$$

We can also estimate the term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{st}$$\end{document} by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert A_{st}\Vert _{\mathscr {C}^{\sigma +\varsigma }_p}\\ &\quad \leqslant {\displaystyle \Vert g\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}^{\sigma }_p}\Vert h\Vert _{C_{T}\mathscr {C}^{\varsigma }}\int _{s}^{t}(T-r)^{-\gamma '}dr+\Vert h\Vert _{C_{T}\mathscr {C}^{\varsigma }}\Vert g\Vert _{C^{\gamma ',1}_{\overline{T},T}\mathscr {C}^{\sigma -\alpha }_p}\int _{s}^{t}(T-r)^{-\gamma '}dr}\\ &\quad \lesssim (T-t)^{-\gamma '}|t-s|\Vert h\Vert _{C_{T}\mathscr {C}^{\varsigma }}\bigg (\Vert g\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}^{\sigma }_p}+\Vert g\Vert _{C^{\gamma ',1}_{\overline{T},T}\mathscr {C}^{\sigma -\alpha }_p}\bigg )\\ &\quad \lesssim \overline{T}^{\gamma -\gamma '}(T-t)^{-\gamma }|t-s|\Vert h\Vert _{C_{T}\mathscr {C}^{\varsigma }}\Vert g\Vert _{\mathscr {L}^{\gamma ',\sigma }_{\overline{T},T,p}}, \end{aligned}$$\end{document}

using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(T-r)^{-\gamma }\leqslant (T-t)^{-\gamma }$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\in [s,t]$$\end{document} . It is left to estimate the term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{st}$$\end{document} that we first rewrite:

$$[eqn]$$ $$[eqn]$$

To estimate the term in line (23), we use Lemma 1 and the estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{1}(t)+I_{2}(t)$$\end{document} from above to obtain

$$[eqn]$$

To estimate the term in line (23), we can also estimate differently using Lemma 1 and an easier estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{1}(t),I_{2}(t)$$\end{document} using the semigroup estimates and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (1-\gamma ')<\alpha $$\end{document} to obtain

$$[eqn]$$

To estimate the term in line (24), we use the commutator for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{t-s}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =-\alpha $$\end{document} and again Lemma 5 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{t,r}=P_{r-t}h_{r}$$\end{document} , yielding

$$[eqn]$$

Applying instead the semigroup commutator for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =-(1-\gamma ')\alpha $$\end{document} yields

$$[eqn]$$

where to bound the time integral, we used that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (1-\gamma ')<\alpha $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\leqslant t$$\end{document} . Together we obtain the desired estimates for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{st}$$\end{document} , which yield together with the estimates for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{st}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{st}$$\end{document} the claim. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Remark 4

The proof of the commutator estimate does not apply if we consider instead of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\in \mathscr {L}_{T,p}^{\gamma ,\sigma }$$\end{document} , a function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\in \mathscr {M}_{T}^{\gamma }\mathscr {C}^{\sigma }\cap C_{T}^{1-\gamma }\mathscr {C}^{\sigma -\alpha }$$\end{document} . The reason is the estimate for the term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{2}$$\end{document} in the above proof, for which we need to employ that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\in C_{T}^{\gamma ,1}\mathscr {C}^{\sigma -\alpha }$$\end{document} .

Remark 5

Assuming less time regularity on g than the Lipschitz assumption above, we can still prove a commutator that gains in time regularity. Indeed, assuming that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\in \mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\sigma }\cap C_{\overline{T},T}^{\gamma ',\theta }\mathscr {C}_{p}^{\sigma -\alpha \theta }\cap C_{\overline{T},T}^{\theta -\gamma '}\mathscr {C}_{p}^{\sigma -\alpha \theta }=:S_{\overline{T},T,p}^{\gamma ',\sigma , \theta }$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in (0,1]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in (0,\alpha \theta )$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma '\in [0,\theta )$$\end{document} , an adaption of the argument yields a commutator estimate of the following form for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [\gamma ',\theta )$$\end{document} ,

$$[eqn]$$

The commutator from Lemma 6 thus corresponds to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =1$$\end{document} . For example, the commutator (25) might be relevant for applying it in the context of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi ^{4}_{3}$$\end{document} equation. In that case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g=J^{T}(Z^{:3:})$$\end{document} for the Wick product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z^{:3:}$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z=J^{T}(\xi )$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} denotes a typical paths of the periodic space-time white noise in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=3$$\end{document} , and one can show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\in C_{T}\mathscr {C}^{1/2-\varepsilon }(\mathbb {T}^{d})\cap C_{T}^{1-\delta }\mathscr {C}^{-3/2-\varepsilon }(\mathbb {T}^{d})$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ,\delta \in (0,1/2)$$\end{document} .

We conclude this section with interpolation estimates for the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{T,p}^{\gamma ,\theta }$$\end{document} .

Lemma 7

(Interpolation estimates) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [0,1)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in [0,\alpha ]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document} . Let moreover \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v\in \mathscr {L}_{T,p}^{\gamma ,\theta }$$\end{document} . Then the following estimates hold true:

It follows that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in (0,\alpha )$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert v\Vert _{C_{T}^{\theta /\alpha }L^{p}}\lesssim \Vert v\Vert _{\mathscr {L}_{T,p}^{0,\theta }}. \end{aligned}$$\end{document}

Furthermore, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\theta }\in [0,\alpha ]$$\end{document} , it holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert v\Vert _{C_{T}^{\gamma ,\tilde{\theta }/\alpha }\mathscr {C}^{\theta -\tilde{\theta }}_p}\lesssim \Vert v\Vert _{\mathscr {L}_{T,p}^{\gamma ,\theta }} \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert v\Vert _{\mathscr {M}_{T}^{\gamma (1-\tilde{\theta }/\alpha )}\mathscr {C}^{\theta -\tilde{\theta }}_{p}}\lesssim \Vert v\Vert _{\mathscr {L}_{T,p}^{\gamma ,\theta }}. \end{aligned}$$\end{document}

If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{T}\in \mathscr {C}^{\theta -\tilde{\theta }}_{p}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\theta }\in [\alpha \gamma ,\alpha ]$$\end{document} , then the following estimate holds true

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert v_{t}\Vert _{\mathscr {C}^{\theta -\tilde{\theta }}_{p}}\lesssim (T-t)^{\tilde{\theta }/\alpha -\gamma }\Vert v\Vert _{\mathscr {L}_{T,p}^{\gamma ,\theta }}+\Vert v_{T}\Vert _{\mathscr {C}^{\theta -\tilde{\theta }}_{p}}. \end{aligned}$$\end{document}

Remark 6

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in (0,1]$$\end{document} and a Banach space X, we recall that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{T}^{0,\theta }X=C_{T}^{\theta }X$$\end{document} .

Proof

To prove (26) we let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\leqslant s\leqslant t\leqslant T$$\end{document} and estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert v_{t}-v_{s}\Vert _{L^{p}}&\leqslant \sum _{j}\Vert \Delta _{j}(v_{t}-v_{s})\Vert _{L^{p}}\\&\lesssim \sum _{j:\,2^{-j}\leqslant |t-s|^{1/\alpha }}2^{-j\theta }\Vert v\Vert _{C_{T}\mathscr {C}_{p}^{\theta }} \\ &\quad +\sum _{j:\,2^{-j}>|t-s|^{1/\alpha }}2^{-j(\theta -\alpha )}|t-s|\Vert v\Vert _{C_{T}^{1}\mathscr {C}_{p}^{\theta -\alpha }}\\&\lesssim |t-s|^{\theta /\alpha }\Vert v\Vert _{C_{T}\mathscr {C}_{p}^{\theta }}+|t-s|^{\theta /\alpha }\Vert v\Vert _{C_{T}^{1}\mathscr {C}_{p}^{\theta -\alpha }}, \end{aligned}$$\end{document}

using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta > 0$$\end{document} for the convergence of the geometric sum and that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta <\alpha $$\end{document} . To prove (27) and (28), we let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\theta }\in [0,\alpha ]$$\end{document} . Then we estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s<t$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Delta _{j}(v_{t}-v_{s})\Vert _{L^{p}}\lesssim (T-t)^{-\gamma }\min \Big (2^{-j\theta }\Vert v\Vert _{\mathscr {M}_{T}^{\gamma }\mathscr {C}_{p}^{\theta }},2^{-j(\theta -\alpha )}|t-s|\Vert v\Vert _{C_{T}^{\gamma ,1}\mathscr {C}_{p}^{\theta -\alpha }}\Big ) \end{aligned}$$\end{document}

and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in [0,T)$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Delta _{j}v_{t}\Vert _{L^{p}}\lesssim \min (2^{-j\theta }(T-t)^{-\gamma }\Vert v\Vert _{\mathscr {M}_{T}^{\gamma }\mathscr {C}_{p}^{\theta }},2^{-j(\theta -\alpha )}\Vert v\Vert _{C_{T}\mathscr {C}_{p}^{\theta -\alpha }}). \end{aligned}$$\end{document}

Thus by interpolation (that is, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min (a,b)\leqslant a^{\varepsilon }b^{1-\varepsilon }$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,b\geqslant 0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \in [0,1]$$\end{document} ) and using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert v\Vert _{C_{T}\mathscr {C}_{p}^{\theta -\alpha }}\lesssim \Vert v\Vert _{\mathscr {L}_{T,p}^{\gamma ,\theta }}$$\end{document} , we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Delta _{j}(v_{t}-v_{s})\Vert _{L^{p}}&\lesssim (T-t)^{-\gamma }2^{-j\theta (1-\tilde{\theta }/\alpha )}2^{-j(\theta -\alpha )\tilde{\theta }/\alpha }|t-s|^{\tilde{\theta }/\alpha }\Vert v\Vert _{\mathscr {L}_{T,p}^{\gamma ,\theta }}\\ &=(T-t)^{-\gamma } 2^{-j(\theta -\tilde{\theta })}|t-s|^{\tilde{\theta }/\alpha }\Vert v\Vert _{\mathscr {L}_{T,p}^{\gamma ,\theta }}, \end{aligned}$$\end{document}

from which (27) follows, and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Delta _{j}v_{t}\Vert _{L^{p}}&\lesssim 2^{-j\theta (1-\tilde{\theta }/\alpha )}(T-t)^{-\gamma (1-\tilde{\theta }/\alpha )}\Vert v\Vert _{\mathscr {M}_{T}^{\gamma }\mathscr {C}_{p}^{\theta }}^{1-\tilde{\theta }/\alpha }\,\, 2^{-j(\theta -\alpha )\tilde{\theta }/\alpha }\Vert v\Vert _{C_{T}\mathscr {C}_{p}^{\theta -\alpha }}^{\tilde{\theta }/\alpha }\\ &\leqslant 2^{-j(\theta -\tilde{\theta })}\Vert v\Vert _{\mathscr {L}_{T,p}^{\gamma ,\theta }}(T-t)^{-\gamma (1-\tilde{\theta }/\alpha )}, \end{aligned}$$\end{document}

which yields (28). Finally, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(T-t)\geqslant 1$$\end{document} , then (29) follows from (28) as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert v_{t}\Vert _{\mathscr {C}^{\theta -\tilde{\theta }}_{p}}\lesssim \Vert v\Vert _{\mathscr {L}_{T,p}^{\gamma ,\theta }}(T-t)^{-\gamma (1-\tilde{\theta }/\alpha )}&\leqslant \Vert v\Vert _{\mathscr {L}_{T,p}^{\gamma ,\theta }}\\ &\leqslant (T-t)^{\tilde{\theta }/\alpha -\gamma } [\Vert v\Vert _{\mathscr {L}_{T,p}^{\gamma ,\theta }}+\Vert v_{T}\Vert _{\mathscr {C}^{\theta -\tilde{\theta }}_{p}}^{1-\tilde{\theta }/\alpha }]\\&\quad +\Vert v_{T}\Vert _{\mathscr {C}^{\theta -\tilde{\theta }}_{p}} \end{aligned}$$\end{document}

using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\theta }/\alpha \geqslant \gamma $$\end{document} . If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(T-t)\leqslant 1$$\end{document} , then (29) follows from

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert v_{t}\Vert _{\mathscr {C}^{\theta -\tilde{\theta }}_{p}}\leqslant \Vert v_{t}-v_{T}\Vert _{\mathscr {C}^{\theta -\tilde{\theta }}_{p}}+\Vert v_{T}\Vert _{\mathscr {C}^{\theta -\tilde{\theta }}_{p}} \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert&\Delta _{j}(v_{t}-v_{T})\Vert _{L^{p}}\\ &\lesssim {\displaystyle \min \Big (2^{-j\theta }(T-t)^{-\gamma }\Vert v\Vert _{\mathscr {M}_{T}^{\gamma }\mathscr {C}_{p}^{\theta }}+2^{-j(\theta -\tilde{\theta })}\Vert v_{T}\Vert _{\mathscr {C}^{\theta -\tilde{\theta }}_{p}}, 2^{-j(\theta -\alpha )}(T-t)^{1-\gamma }\Vert v\Vert _{C_{T}^{1-\gamma }\mathscr {C}_{p}^{\theta -\alpha }}\Big )} \\ &\leqslant {\displaystyle \min \Big (2^{-j\theta }(T-t)^{-\gamma }\Vert v\Vert _{\mathscr {M}_{T}^{\gamma }\mathscr {C}_{p}^{\theta }}, 2^{-j(\theta -\alpha )}(T-t)^{1-\gamma }\Vert v\Vert _{C_{T}^{1-\gamma }\mathscr {C}_{p}^{\theta -\alpha }}\Big )+2^{-j(\theta -\tilde{\theta })}\Vert v_{T}\Vert _{\mathscr {C}^{\theta -\tilde{\theta }}_{p}}.} \end{aligned}$$\end{document}

By interpolation as above, we thus have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Delta _{j}(v_{t}-v_{T})\Vert _{L^{p}}\lesssim 2^{-j(\theta -\tilde{\theta })}(T-t)^{\tilde{\theta }/\alpha -\gamma }\Vert v\Vert _{\mathscr {L}_{T,p}^{\gamma ,\theta }}, \end{aligned}$$\end{document}

such that together (29) follows. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Solving the Kolmogorov Backward Equation

In this section, we develop a concise solution theory that simultaneously treats singular and non-singular terminal condition for the Kolmogorov backward equation.

We start by solving the Kolmogorov equation in the Young regime, that is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta >(1-\alpha )/2$$\end{document} .

Theorem 2

2 Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (\frac{1-\alpha }{2}, 0)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\in C_T\mathscr {C}^{\beta }_{\mathbb {R}^{d}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C_T\mathscr {C}^{\beta }_{p}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}\in \mathscr {C}^{\alpha +\beta }_{p}$$\end{document} . Then the PDE

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _{t}u=\mathfrak {L}^{\alpha }_{\nu }u-V\cdot \nabla u+f,\quad u(T,\cdot )=u^{T}, \end{aligned}$$\end{document}

has a unique mild solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in C_{T}\mathscr {C}^{\alpha +\beta }\cap C_{T}^{1}\mathscr {C}^{\beta }$$\end{document} (i.p. by (26), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in C_{T}^{(\alpha +\beta )/\alpha }L^{p}$$\end{document} ). Moreover, the solution map

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathscr {C}^{\alpha +\beta }_{p} \times C_T\mathscr {C}^{\beta }_{p} \times C_T\mathscr {C}^{\beta }_{\mathbb {R}^{d}} \ni (u^{T},f,V)\mapsto u \in \mathscr {L}_{T,p}^{0,\alpha +\beta } \end{aligned}$$\end{document}

is continuous.

Furthermore, for a singular terminal conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}\in \mathscr {C}^{(1-\gamma )\alpha +\beta }_{p}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [0,1)$$\end{document} , the solution u is obtained in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{T,p}^{\gamma ,\alpha +\beta }$$\end{document} .

Proof

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}\in \mathscr {C}^{\alpha +\beta }_{p}$$\end{document} . We first prove that the solution exists in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{T,p}^{\gamma ,\alpha +\beta }$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} . Afterwards we argue that indeed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \mathscr {L}_{T,p}^{0,\alpha +\beta }$$\end{document} . The proof follows from the Banach fixed point theorem applied to the map

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\mathscr {L}_{\overline{T},T,p}^{\gamma ,\alpha +\beta }\ni u\mapsto \Phi ^{{\overline{T}}, T} (u)\in \mathscr {L}_{\overline{T},T,p}^{\gamma ,\alpha +\beta }\text { with } \Phi ^{{\overline{T}}, T}u(t) \\ &\qquad =P_{T-t}u^{T}+J^{T}(\nabla u\cdot V-f)(t), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^{T}(v)(t)=\int _{t}^{T}P_{r-t}v(r)dr$$\end{document} . We show below that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}\in (0,T]$$\end{document} small enough, the map is a contraction. By the Schauder estimates (Corollary 1), we obtain that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\mapsto P_{T-t}u^{T}\in \mathscr {L}_{\overline{T},T,p}^{0,\alpha +\beta }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^{T}(f)\in \mathscr {L}_{\overline{T},T,p}^{0,\alpha +\beta }$$\end{document} . Furthermore, the Schauder estimates (Corollary 1) and the interpolation estimate (28) from Lemma 7 yield that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma '\in (0,\gamma )$$\end{document} chosen, such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =\gamma '(1-\theta /\alpha )$$\end{document} for a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in (0,\alpha +2\beta -1)$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert J^{T}(\nabla u\cdot V)\Vert _{\mathscr {L}_{\overline{T},T,p}^{\gamma ,\alpha +\beta }}&\lesssim \overline{T}^{\gamma -\gamma '}\Vert \nabla u\cdot V\Vert _{\mathscr {M}_{\overline{T},T}^{\gamma '}\mathscr {C}^{\beta }_{p}}\\ &\lesssim \overline{T}^{\gamma -\gamma '}\Vert \nabla u\Vert _{\mathscr {M}_{\overline{T},T}^{\gamma '}(\mathscr {C}^{\alpha +\beta -1-\theta }_{p})^{d}}\Vert V\Vert _{C_{T}\mathscr {C}^{\beta }_{\mathbb {R}^{d}}}\\ &\lesssim \overline{T}^{\gamma -\gamma '}\Vert u\Vert _{\mathscr {L}_{\overline{T},T,p}^{\gamma ,\alpha +\beta }}\Vert V\Vert _{C_{T}\mathscr {C}^{\beta }_{\mathbb {R}^{d}}}. \end{aligned}$$\end{document}

Notice that due to the choice of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} the regularity of the resonant product $[eqn]$ is strictly positive. Thus, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}\in (0,T]$$\end{document} sufficiently small, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi ^{{\overline{T}}, T}$$\end{document} is a contraction on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{\overline{T},T,p}^{\gamma ,\alpha +\beta }$$\end{document} and we obtain a solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \mathscr {L}_{\overline{T},T,p}^{\gamma ,\alpha +\beta }$$\end{document} (i.e. the fixed point of the map).

By plugging the solution back in the contraction map and using the interpolation estimate (29) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =\alpha +\beta , \tilde{\theta }=\gamma \alpha $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,(\alpha +2\beta -1)/\alpha )$$\end{document} , we then obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u\Vert _{\mathscr {L}_{\overline{T},T,p}^{0,\alpha +\beta }}&=\Vert \Phi ^{\overline{T},T}(u)\Vert _{\mathscr {L}_{\overline{T},T,p}^{0,\alpha +\beta }}\nonumber \\ &\lesssim \Vert P_{T-\cdot }u^{T}+J^{T}(f)\Vert _{\mathscr {L}_{\overline{T},T,p}^{0,\alpha +\beta }}+\Vert u\Vert _{C_{\overline{T},T}\mathscr {C}^{\alpha +\beta -\gamma \alpha }}\Vert V\Vert _{C_{T}\mathscr {C}^{\beta }_{\mathbb {R}^{d}}}\nonumber \\ &\lesssim \Vert P_{T-\cdot }u^{T}+J^{T}(f)\Vert _{\mathscr {L}_{T,p}^{0,\alpha +\beta }}+[\Vert u\Vert _{\mathscr {L}_{\overline{T},T,p}^{\gamma ,\alpha +\beta }}+\Vert u_{T}\Vert _{\mathscr {C}^{\alpha +\beta }_{p}}]\Vert V\Vert _{C_{T}\mathscr {C}^{\beta }_{\mathbb {R}^{d}}}. \end{aligned}$$\end{document}

This implies that indeed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \mathscr {L}^{0,\alpha +\beta }_{\overline{T},T,p}$$\end{document} and we constructed the solution on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-\overline{T},T]$$\end{document} .

Moreover, the choice of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{T}}$$\end{document} does not depend on the terminal condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^T$$\end{document} and therefore we can iterate the construction of the solution on subintervals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-k\overline{T},T-(k-1)\overline{T}]$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\in {1,\dots ,n}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathbb {N}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T-n\overline{T}\leqslant 0$$\end{document} . Here, we choose the terminal condition of the solution on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-k\overline{T},T-(k-1)\overline{T}]$$\end{document} equal to the initial value of the solution constructed in the previous iteration step. We then obtain the solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \mathscr {L}_{T,p}^{0,\alpha +\beta }$$\end{document} on [0, T] by patching the solutions on the subintervals together. Indeed, u is the fixed point of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi ^{T,T}$$\end{document} , due to the semigroup property \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{t}P_{s}=P_{t+s}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t,s\geqslant 0$$\end{document} .

The continuity of the solution map follows from

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u\Vert _{\mathscr {L}_{T,p}^{0,\alpha +\beta }}\leqslant \sum _{k=0}^{n}\Vert u\Vert _{\mathscr {L}_{\overline{T},T-k\overline{T},p}^{0,\alpha +\beta }} \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathbb {N}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T-(n+1)\overline{T}\leqslant 0$$\end{document} , together with (31) applied for each of the terms on the right-hand side and the contraction property on each of the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{\overline{T},T-k\overline{T},p}^{\gamma ,\alpha +\beta }$$\end{document} . For a terminal condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}\in \mathscr {C}^{(1-\gamma )\alpha +\beta }_{p}$$\end{document} , the above arguments show that we obtain a solution in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{T,p}^{\gamma ,\alpha +\beta }$$\end{document} . Notice that the blow-up just occurs for the solution on the last subinterval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-\overline{T},T]$$\end{document} . That is, the solutions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-k\overline{T},T-(k-1)\overline{T}]$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2,\dots ,n$$\end{document} have a regular terminal condition in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {C}^{\alpha +\beta }_{p}$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Next, we define the space of enhanced distributions and afterwards the solution space for solving the generator equation with paracontrolled terminal condition and right-hand side in the rough regime \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \leqslant \frac{1-\alpha }{2}$$\end{document} . For that, we define for a Banach space X, the blow-up space

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathscr {M}_{\mathring{\Delta }_{T}}^{\gamma }X=\{g:\mathring{\Delta }_{T}\rightarrow X\mid \sup _{0\leqslant s<t\leqslant T}(t-s)^{\gamma }\Vert g(s,t)\Vert _{X}<\infty \} \end{aligned}$$\end{document}

for the triangle without diagonal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathring{\Delta }_{T}:=\{(s,t)\in [0,T]^{2}\mid s<t\}$$\end{document} . Below we take $[eqn]$ for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta \in C_T C^{\infty }_{b}(\mathbb {R}^{d},\mathbb {R}^{d})$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i,j\in \{1,...,d\}$$\end{document} .

Definition 2

(Enhanced drift) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T>0$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (\frac{2-2\alpha }{3},\!\frac{1-\alpha }{2}]$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [\frac{2\beta +2\alpha -1}{\alpha },1)$$\end{document} , we define the space of enhanced drifts \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {X}^{\beta ,\gamma }$$\end{document} as the closure of

$$[eqn]$$

in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_T\mathscr {C}^{\beta +(1-\gamma )\alpha }_{\mathbb {R}^{d}}\times \mathscr {M}_{\mathring{\Delta }_{T}}^{\gamma }\mathscr {C}^{2\beta +\alpha -1}_{\mathbb {R}^{d\times d}}$$\end{document} . We say that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}$$\end{document} is a lift or an enhancement of V if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}_{1}=V$$\end{document} and we also write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\in \mathscr {X}^{\beta ,\gamma }$$\end{document} identifying V with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathscr {V}_{1},\mathscr {V}_{2})$$\end{document} .

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (\frac{1-\alpha }{2},0)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [\frac{\beta -1}{\alpha },1)$$\end{document} , we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {X}^{\beta ,\gamma }=C_{T}\mathscr {C}^{\beta +(1-\gamma )\alpha }$$\end{document} .

Remark 7

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}\in \mathscr {X}^{\beta ,\gamma }$$\end{document} , we assume on the first component \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}_{1}\in C_{T}\mathscr {C}^{\beta +\alpha (1-\gamma )}$$\end{document} . We think of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \sim 1$$\end{document} , that is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma <1$$\end{document} , but very close to 1. The assumptions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}$$\end{document} in particular imply by the semigroup estimates that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\mapsto P_{T-t}V_{T}^{i}\in \mathscr {M}_{T}^{\gamma }\mathscr {C}^{\alpha +\beta }$$\end{document} . Furthermore, from $[eqn]$ follows that $[eqn]$ . Indeed, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma <1$$\end{document} , we can estimate

$$[eqn]$$

using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma <1$$\end{document} . Analogously we obtain that $[eqn]$ with a uniform bound in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\in (0,T]$$\end{document} . The assumptions on the enhancement will become handy, as soon as we consider paracontrolled solutions on subintervals of [0, T].

Remark 8

We assume the lower bound on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} to ensure that the regularity of V, respectively, and the regularity of the resonant products $[eqn]$ are negative. That is, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma <(2\beta +2\alpha -1)/\alpha $$\end{document} , we obtain that $[eqn]$ due to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\in C_{T}\mathscr {C}^{\beta +(1-\gamma )\alpha }$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\beta +(2-\gamma )\alpha -1\geqslant 0$$\end{document} . In this case, V has enough regularity, so that the Kolmogorov PDE can be solved with the classical approach. We exclude this case here, as we explicitly treat the singular case.

Example 1

Examples for enhanced drifts are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V=\xi (\omega )$$\end{document} for a typical paths \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi (\omega )$$\end{document} of a periodic spatial white noise \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \in \mathscr {C}^{\beta }(\mathbb {T})$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =1/2-\varepsilon $$\end{document} and any small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} or, in multidimensions, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V=\nabla W (\omega )$$\end{document} for the periodic Brownian sheet W. These examples were covered already in [2, Lemma 5.1 and Remark 5.5]; however, the enhancement assumption was less restrictive there. Nonetheless, it can be shown that those drifts can be enhanced in the above sense for almost all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} . The argument is analogue to [2] and was carried out in [17, Lemma 4.45 and Remark 4.49].

Definition 3

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2]$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (\frac{2-2\alpha }{3},\frac{1-\alpha }{2}]$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T>0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\in {\mathscr {X}}^{\beta ,\gamma '}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma '\in [\frac{2\beta +2\alpha -1}{\alpha },1)$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\prime }\in \mathscr {C}^{\alpha +\beta -1}_p$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (\gamma ',\frac{\alpha }{2-\alpha -3\beta }\gamma ')$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}\in (0,T]$$\end{document} , we define the space of paracontrolled distributions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {D}_{\overline{T},T}^{\gamma } = \mathscr {D}_{\overline{T},T}^{\gamma ,\gamma '}(\mathscr {V}, u^{T,\prime })$$\end{document} as the set of tuples \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u,u')\in \mathscr {L}_{\overline{T},T,p}^{\gamma ',\alpha +\beta }\times (\mathscr {L}_{\overline{T},T,p}^{\gamma ,\alpha +\beta -1})^{d}$$\end{document} , such that

$$[eqn]$$

We define a metric on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {D}_{\overline{T},T}^{\gamma }$$\end{document} by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} d_{\mathscr {D}_{\overline{T},T}^{\gamma }}((u,u'),(v,v'))&:=\Vert u-v\Vert _{\mathscr {D}_{\overline{T},T}^{\gamma }} \\&:= \Vert u-v\Vert _{\mathscr {L}_{\overline{T},T,p}^{\gamma ',\alpha +\beta }} +\Vert u'-v'\Vert _{(\mathscr {L}_{\overline{T},T,p}^{\gamma ,\alpha +\beta -1})^{d}}\\ &\quad +\Vert u^{\sharp }-v^{\sharp }\Vert _{\mathscr {L}_{\overline{T},T,p}^{\gamma ,2(\alpha +\beta )-1}}. \end{aligned}$$\end{document}

Then, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathscr {D}_{\overline{T},T}^{\gamma },d_{\mathscr {D}_{\overline{T},T}^{\gamma }})$$\end{document} is a complete metric space. If moreover \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v,v') \in \mathscr {D}_{\overline{T},T}^{\gamma ,\gamma '}(\mathscr {W}, v^{T,\prime })$$\end{document} for different data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathscr {W},v^{T,\prime })\in \mathscr {X}^{\beta ,\gamma '}\times \mathscr {C}^{\alpha +\beta -1}_p$$\end{document} , then we use the same definition for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert u-v\Vert _{\mathscr {D}_{\overline{T},T}^{\gamma }}$$\end{document} , despite the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u,u')$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v,v')$$\end{document} do not live in the same space.

Remark 9

The intuition behind the paracontrolled ansatz is as follows. Assume for simplicity regular data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u^{T},f)\in \mathscr {C}^{2(\alpha +\beta )-1}_p\times \mathscr {L}_{T,p}^{0,\alpha +2\beta -1}$$\end{document} . Assume also that we found a solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \mathscr {L}_{T,p}^{0,\alpha +\beta }$$\end{document} and that we can make sense of the resonant product $[eqn]$ in such a way that it has its natural regularity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{T}\mathscr {C}^{2\beta +\alpha -1}_p$$\end{document} , despite the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\beta +\alpha -1\leqslant 0$$\end{document} . Then we would get that

$$[eqn]$$

is more regular than u. Indeed, by the Schauder estimates for the first four terms and by the commutator estimate from Lemma 6, we obtain that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{\sharp }\in \mathscr {L}_{T,p}^{0,2(\alpha +\beta )-1}$$\end{document} . This explains why the paracontrolled ansatz might be justified. The reason why the ansatz is useful is that it isolates the singular part of u in a paraproduct that we can handle by commutator estimates and the assumptions on V.

Our main theorem of this section is the following. We give its proof after the corollary below.

Theorem 3

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T>0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (\frac{2-2\alpha }{3},\frac{1-\alpha }{2}]$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}\in \mathscr {X}^{\beta ,\gamma '}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma '\in [\frac{2\beta +2\alpha -1}{\alpha },1)$$\end{document} . Let

$$[eqn]$$

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\sharp }\in \mathscr {L}_{T,p}^{\gamma ',\alpha +2\beta -1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f'\in (\mathscr {L}_{T,p}^{\gamma ',\alpha +\beta -1})^{d}$$\end{document} and

$$[eqn]$$

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\sharp }\in \mathscr {C}^{(2-\gamma ')\alpha +2\beta -1}_p$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\prime }\in (\mathscr {C}^{\alpha +\beta -1}_p)^{d}$$\end{document} .

Then for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (\gamma ',\frac{\alpha }{2-\alpha -3\beta }\gamma ')$$\end{document} there exists a unique mild solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u,u^{\prime })\in \mathscr {D}_{T}^{\gamma }(\mathscr {V}, u^{T,\prime })$$\end{document} of the singular Kolmogorov backward PDE

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathscr {G}^{\mathscr {V}}u=(\partial _{t}-\mathfrak {L}^{\alpha }_{\nu }+\mathscr {V}\cdot \nabla )u=f,\qquad u(T,\cdot )=u^{T}. \end{aligned}$$\end{document}

In particular, the product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}\cdot \nabla u\in \mathscr {M}_{T}^{\gamma }\mathscr {C}^{\beta }$$\end{document} is well-defined.

Remark 10

As \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{T,p}^{\tilde{\gamma },\theta }\subset \mathscr {L}_{T,p}^{\gamma ',\theta }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {C}_{p}^{(2-\tilde{\gamma })\alpha +2\beta -1}\subset \mathscr {C}_{p}^{(2-\gamma ')\alpha +2\beta -1}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\gamma }\in [0,\gamma ']$$\end{document} , we can in particular treat \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\sharp }\in \mathscr {L}_{T,p}^{\tilde{\gamma },\alpha +2\beta -1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\prime }\in \mathscr {L}_{T,p}^{\tilde{\gamma },\alpha +\beta -1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\sharp }\in \mathscr {C}_{p}^{(2-\tilde{\gamma })\alpha +2\beta -1}$$\end{document} .

Example 2

Examples for right-hand sides and terminal conditions, which are paracontrolled by V, respectively, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{T}$$\end{document} , are the following. Clearly we can take as a right-hand side \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=V^{i}$$\end{document} , i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f'=e_{i}$$\end{document} for the i-th unit vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e_{i}$$\end{document} . Another example would be \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=J^{T}(\nabla V^{i})\cdot V$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\in \{1,\dots ,d\}$$\end{document} , where $[eqn]$ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\prime }=J^{T}(\nabla V^{i})$$\end{document} . Considering such singular right-hand sides is relevant to define additive functionals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{0}^{t}f(s,X_{s})ds$$\end{document} of the Markov process X solving the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {G}^{\mathscr {V}}$$\end{document} -martingale problem in [17, section 4].

Similar, as a terminal condition, we can take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}=V^{i}_{T}$$\end{document} , i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\prime }=e_{i}$$\end{document} . More interestingly, in the case of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\prime }=0$$\end{document} , the terminal condition can still be irregular, but is such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\mapsto P_{T-t}u^{T}=P_{T-t}u^{T,\sharp }\in \mathscr {M}_{T}^{\gamma '}\mathscr {C}^{2(\alpha +\beta )-1}_p$$\end{document} . As \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{2\alpha +2\beta -1}{\alpha }\leqslant \gamma '$$\end{document} and thus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2-\gamma ')\alpha +2\beta -1\leqslant 0$$\end{document} , another example for a terminal condition, that can be treated with our approach would be a distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}=u^{T,\sharp }\in \mathscr {C}^{0}_p$$\end{document} . An example would be \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}=\delta _{0}\in \mathscr {C}^{0}_{1}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _{0}$$\end{document} denotes the Dirac measure at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=0$$\end{document} . In particular, the Fokker-Planck equation with initial condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _{0}$$\end{document} corresponding to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {G}^{\mathscr {V}}$$\end{document} can be solved (cf. [17, Theorem 5.4]). More generally, terminal conditions equal to fractional derivatives of the Dirac measure can be considered.

Moreover, in the setting of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\in \mathscr {C}^{\beta }(\mathbb {T}^d)$$\end{document} (periodic, time-independent distribution), right-hand sides \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=\Delta g(\Delta ^{-1}(V))$$\end{document} for smooth functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g:\mathbb {R}^{d}\rightarrow \mathbb {R}^{d}$$\end{document} can be shown to be paracontrolled by V in the above sense.

In the case of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\prime }=0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\sharp }\in \mathscr {C}^{2(\alpha +\beta )-1}$$\end{document} , the terminal condition is sufficiently regular, such that we can prove that the solution of the equation is an element of the solution space without blow-up (provided that f admits zero blow-up). We define, in the case of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\prime }=0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\sharp }\in \mathscr {C}^{2(\alpha +\beta )-1}$$\end{document} , the paracontrolled solution space as

$$[eqn]$$

Corollary 2

(Regular terminal condition) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T>0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (\frac{2-2\alpha }{3},\frac{1-\alpha }{2}]$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}\in \mathscr {X}^{\beta ,\gamma '}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma '\in [\frac{2\beta +2\alpha -1}{\alpha },1)$$\end{document} . Let $[eqn]$ for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\sharp }\in \mathscr {L}_{T,p}^{0,\alpha +2\beta -1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\prime }\in \mathscr {L}_{T,p}^{0,\alpha +\beta -1}$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}=u^{T,\sharp }\in \mathscr {C}^{2\alpha +2\beta -1}_p$$\end{document} be non-singular.

Then, there exists a unique mild solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in D_{T}$$\end{document} of the generator equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathscr {G}^{\mathscr {V}}u=f,\qquad u(T,\cdot )=u^{T}. \end{aligned}$$\end{document}

The proof is deferred to page 29.

Remark 11

The proof of the corollary only uses that $[eqn]$ , which is implied by the stronger assumption \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}\in \mathscr {X}^{\beta ,\gamma '}$$\end{document} (cf. Remark 7).

Proof

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}\in \mathscr {X}^{\beta ,\gamma '}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}_{1}=V$$\end{document} , $[eqn]$ for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma '\in [\frac{2\beta +2\alpha -1}{\alpha },1)$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}\in (0,T]$$\end{document} to be chosen later and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (\gamma ',\frac{\alpha }{2-\alpha -3\beta }\gamma ')$$\end{document} . Then we define the contraction mapping as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \phi =\phi ^{\overline{T},T}:\mathscr {D}^{\gamma }_{\overline{T},T}\rightarrow \mathscr {D}^{\gamma }_{\overline{T},T},\quad (u,u^{\prime })\mapsto (\psi (u),\nabla u-f^{\prime }) \end{aligned}$$\end{document}

for

$$[eqn]$$

where we define

$$[eqn]$$ $$[eqn]$$

with $[eqn]$ . The commutators are defined as follows:

$$[eqn]$$

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{1}$$\end{document} denotes the commutator on the semigroup \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{T-\cdot }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{2}$$\end{document} is the commutator from Lemma 6. Furthermore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{3}$$\end{document} denotes the commutator from [15, Lemma 2.4], that is

$$[eqn]$$

For the terms in (34), we obtain with Remark 7, the paraproduct estimates and [15, Lemma 2.4] using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\beta +2\alpha -2>0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\beta +\alpha -1\leqslant 0$$\end{document} ,

$$[eqn]$$

For the terms in (35), we have by the estimate on the paraproduct and the definition of the enhanced distribution space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {X}^{\beta ,\gamma '}$$\end{document}

$$[eqn]$$

where we used that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\alpha +3\beta -2>0$$\end{document} . By the commutator estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{3}$$\end{document} from [15, Lemma 2.4] and the estimates for the semigroup to control \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{T-\cdot }\nabla V_{T}$$\end{document} , we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert C_{3}(u^{T,\prime },P_{T-\cdot }\partial _{i} V_{T},V^{i})\Vert _{\mathscr {M}_{\overline{T},T}^{\gamma '}\mathscr {C}_{p}^{3\beta +2\alpha -2}}\lesssim \Vert u^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^d}\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}^{2}, \end{aligned}$$\end{document}

using again \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\alpha +3\beta -2>0$$\end{document} by the assumption on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} .

Define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon :=\alpha -\alpha \frac{\gamma '}{\gamma }$$\end{document} . Then it follows that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \in (0,3\beta +2\alpha -2)$$\end{document} by the assumption on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} . Subtracting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} regularity for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{\prime }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{\sharp }$$\end{document} , we can estimate the resonant product along the same lines as above, due to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\beta +2\alpha -2-\varepsilon >0$$\end{document} , obtaining

$$[eqn]$$

In (36), we moreover used the interpolation bound (28) for the norm of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{\prime }$$\end{document} , that is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u^{\prime }\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}_{p}^{\alpha +\beta -1-\varepsilon }}=\Vert u^{\prime }\Vert _{\mathscr {M}^{\gamma (1-\varepsilon /\alpha )}_{\overline{T},T}\mathscr {C}_{p}^{\alpha +\beta -1-\varepsilon }}\lesssim \Vert u^{\prime }\Vert _{\mathscr {L}^{\gamma ,\alpha +\beta -1}_{\overline{T},T,p}} \end{aligned}$$\end{document}

by the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} , and analogously for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{\sharp }$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u,u^{\prime }),(v,v^{\prime })\in \mathscr {D}_{\overline{T},T}^{\gamma }(\mathscr {V}, u^{T,\prime })$$\end{document} , this also implies the Lipschitz bound:

$$[eqn]$$

Next, we show that indeed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (u,u^{\prime })=(\psi (u),\nabla u-f^{\prime })\in \mathscr {D}^{\gamma }_{\overline{T},T}$$\end{document} and that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} is a contraction for small enough \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}$$\end{document} .

Towards the first aim, we note that by (33),

$$[eqn]$$

By the Schauder estimates, we obtain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{T-\cdot }u^{T,\sharp }+J^{T}(f^{\sharp })\in \mathscr {L}_{\overline{T},T,p}^{\gamma ',2\alpha +2\beta -1}$$\end{document} and

$$[eqn]$$

using the estimate for the resonant product from above. Utilizing the commutator estimate (Lemma 6), we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert C_{2}(\nabla u,V)\Vert _{\mathscr {L}^{\gamma ,2(\alpha +\beta )-1}_{\overline{T},T,p}}&\lesssim \overline{T}^{\gamma -\gamma '}\Vert V\Vert _{C_{T}\mathscr {C}^{\beta }_{\mathbb {R}^{d}}}\Vert u\Vert _{\mathscr {L}_{\overline{T},T,p}^{\gamma ',\alpha +\beta }}\\&\lesssim \overline{T}^{\gamma -\gamma '}\Vert V\Vert _{C_{T}\mathscr {C}^{\beta }_{\mathbb {R}^{d}}}\Vert (u,u^{\prime })\Vert _{\mathscr {D}_{\overline{T},T}^{\gamma }}. \end{aligned}$$\end{document}

By \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{T}\in \mathscr {C}^{\beta +(1-\gamma ')\alpha }_{\mathbb {R}^{d}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\prime }\in (\mathscr {C}_{p}^{\alpha +\beta -1})^d$$\end{document} and the commutator estimate 3 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{1}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =\gamma '\alpha $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha +\beta -1\in (0,1)$$\end{document} and again Lemma 6 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{2}$$\end{document} , we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert C_{1}(u^{T,\prime },V_{T})+C_{2}(f^{\prime },V)\Vert _{\mathscr {L}^{\gamma ',2(\alpha +\beta )-1}_{\overline{T},T,p}}\\ &\quad \lesssim \Vert V\Vert _{C_{T}\mathscr {C}^{\beta +(1-\gamma ')\alpha }_{\mathbb {R}^{d}}}(\Vert u^{T,\prime }\Vert _{(\mathscr {C}_{p}^{\alpha +\beta -1})^d}+\Vert f^{\prime }\Vert _{\mathscr {L}_{T,p}^{\gamma ',\alpha +\beta -1}}). \end{aligned}$$\end{document}

Hence, together we obtain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (u,u^{\prime })^{\sharp }\in \mathscr {L}^{\gamma ,2\alpha +2\beta -1}_{\overline{T},T,p}$$\end{document} .

Next, we show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (u)\in \mathscr {L}^{\gamma ',\alpha +\beta }_{\overline{T},T,p}$$\end{document} .

Define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma '':=\gamma '(1-\varepsilon _{1}/\alpha )$$\end{document} for a fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _{1}\in (0,(\alpha +\beta -1)\wedge \frac{(1-\gamma ')\alpha }{2-\alpha -3\beta })=(0,\frac{(1-\gamma ')\alpha }{2-\alpha -3\beta })$$\end{document} and define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _{2}:=\alpha -\alpha \frac{\gamma ''}{\gamma }$$\end{document} . Then it follows that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _{2}\in (0,3\beta +2\alpha -2+(1-\gamma ')\alpha )$$\end{document} . Using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\in C_{T}\mathscr {C}^{\beta +(1-\gamma ')\alpha }_{\mathbb {R}^{d}}$$\end{document} and applying twice the interpolation bound (28) (once for u and once for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{\sharp }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{\prime }$$\end{document} ), an analogue estimate as for the resonant product $[eqn]$ yields that

$$[eqn]$$

Thus, we obtain that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert \psi (u)\Vert _{\mathscr {L}^{\gamma ',\alpha +\beta }_{\overline{T},T,p}}\\ &\quad =\Vert P_{T-\cdot }u^{T}+J^{T}(f)+J^{T}(\nabla u\cdot \mathscr {V})\Vert _{\mathscr {L}^{\gamma ',\alpha +\beta }_{\overline{T},T}}\\ &\quad \leqslant \Vert P_{T-\cdot }u^{T}\Vert _{\mathscr {L}^{\gamma ',\alpha +\beta }_{\overline{T},T}}+\Vert f\Vert _{\mathscr {L}_{T,p}^{\gamma ',\beta }}+\Vert J^{T}(\nabla u\cdot \mathscr {V})\Vert _{\mathscr {L}^{\gamma ',\alpha +\beta }_{\overline{T},T}}\\ &\quad \lesssim {\displaystyle \Vert u^{T,\sharp }\Vert _{\mathscr {C}^{(2-\gamma ')\alpha +2\beta -1}_p}+\Vert u^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^d}\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}+\Vert C_{1}(u^{T,\prime },V_{T})\Vert _{\mathscr {M}_{T}^{\gamma '}\mathscr {C}_{p}^{2\alpha +2\beta -1}}}\\ &\qquad +\Vert f\Vert _{\mathscr {L}_{T,p}^{\gamma ',\beta }}+\overline{T}^{\gamma '-\gamma ''}\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}(1+\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}})\Vert u\Vert _{\mathscr {D}_{\overline{T},T}^{\gamma ,\alpha +\beta }}, \end{aligned}$$\end{document}

which yields in particular \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (u)\in \mathscr {L}^{\gamma ',\alpha +\beta }_{\overline{T},T}$$\end{document} . We estimate the Gubinelli derivative \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (u,u^{\prime })^{\prime }=\nabla u-f^{\prime }$$\end{document} as follows

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert \nabla u-f^{\prime }\Vert _{(\mathscr {L}^{\gamma ,\alpha +\beta -1}_{\overline{T},T,p})^d}\\ &\quad \lesssim \Vert \nabla u\Vert _{\mathscr {M}^{\gamma }_{\overline{T},T}(\mathscr {C}_{p}^{\alpha +\beta -1})^d}+\Vert \nabla u\Vert _{C^{1-\gamma }_{\overline{T},T}(\mathscr {C}_{p}^{\beta -1})^d}+\Vert \nabla u\Vert _{C^{\gamma ,1}_{\overline{T},T}(\mathscr {C}_{p}^{\beta -1})^d}+\Vert f^{\prime }\Vert _{(\mathscr {L}^{\gamma ,\alpha +\beta -1}_{\overline{T},T,p})^d} \\ &\quad \lesssim \overline{T}^{\gamma -\gamma '}\big (\Vert u\Vert _{\mathscr {L}^{\gamma ',\alpha +\beta }_{\overline{T},T,p}}+\Vert f^{\prime }\Vert _{(\mathscr {L}^{\gamma ',\alpha +\beta -1}_{\overline{T},T,p})^d}\big )\\ &\quad \lesssim \overline{T}^{\gamma -\gamma '}\big (\Vert u\Vert _{\mathscr {D}^{\gamma }_{\overline{T},T}}+\Vert f^{\prime }\Vert _{(\mathscr {L}^{\gamma ',\alpha +\beta -1}_{\overline{T},T,p})^d}\big ), \end{aligned}$$\end{document}

where we exploit the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma -\gamma '>0$$\end{document} to obtain a non-trivial factor depending on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}$$\end{document} . Together with the estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (u)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (u,u^{\prime })^{\sharp }$$\end{document} , this yields \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (u,u^{\prime })=(\psi (u),\nabla u-f^{\prime })\in \mathscr {D}_{\overline{T},T}^{\gamma }$$\end{document} .

The contraction property follows using the above estimates for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (u)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (u,u^{\prime })^{\sharp }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (u,u^{\prime })^{\prime }$$\end{document} , utilizing linearity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} (for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}=0, f=0$$\end{document} ), such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert (\psi (u),\nabla u-f^{\prime })-(\psi (v),\nabla v-f^{\prime })\Vert _{\mathscr {D}^{\gamma }_{\overline{T},T}}\nonumber \\&\quad = \Vert \psi (u)-\psi (v)\Vert _{\mathscr {L}^{\gamma ',\alpha +\beta }_{\overline{T},T,p}}+\Vert \nabla u-\nabla v\Vert _{(\mathscr {L}^{\gamma ,\alpha +\beta -1}_{\overline{T},T,p})^d}\nonumber \\&\qquad +\Vert \phi (u,u^{\prime })^{\sharp }-\phi (v,v^{\prime })^{\sharp }\Vert _{\mathscr {L}^{\gamma ,2\alpha +2\beta -1}_{\overline{T},T,p}} \nonumber \\&\quad \lesssim (\overline{T}^{\gamma -\gamma '}\vee \overline{T}^{\gamma '-\gamma ''})\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}(1+\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}})\Vert (u,u^{\prime })-(v,v^{\prime })\Vert _{\mathscr {D}_{\overline{T},T}^{\gamma }}. \end{aligned}$$\end{document}

Now, we can choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}$$\end{document} small enough, such that the implicit constant times the factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\overline{T}^{\gamma -\gamma '}\vee \overline{T}^{\gamma '-\gamma ''})\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}(1+\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}})$$\end{document} is strictly less than 1, such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi =\phi ^{\overline{T},T}$$\end{document} is a contraction on the corresponding space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {D}_{\overline{T},T}^{\gamma }$$\end{document} . It is left to show that we can obtain a paracontrolled solution in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {D}_{T}^{\gamma }$$\end{document} on the whole interval [0, T]. The solution on [0, T] is obtained by patching the solutions on the subintervals of length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}$$\end{document} together. Indeed, let inductively \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{[T-\overline{T},T]}$$\end{document} be the solution on the subinterval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-\overline{T},T]$$\end{document} with terminal condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{[T-k\overline{T},T-(k-1)\overline{T}]}$$\end{document} be the solution on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-k\overline{T},T-(k-1)\overline{T}]$$\end{document} with terminal condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{[T-k\overline{T},T-(k-1)\overline{T}]}_{T-(k-1)\overline{T}}=u^{[T-(k-1)\overline{T},T-(k-2)\overline{T}]}_{T-(k-1)\overline{T}}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2,\dots , n$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathbb {N}$$\end{document} , such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T-n\overline{T}\leqslant 0$$\end{document} . There is a small subtlety, as we consider the solution on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-k\overline{T},T-(k-1)\overline{T}]$$\end{document} , that is paracontrolled by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^{T}(V)$$\end{document} (and not by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^{T-(k-1)\overline{T}}(V)$$\end{document} ). That is, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2,\dots ,n$$\end{document} , the solution has the paracontrolled structure,

$$[eqn]$$

Notice that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\geqslant 2$$\end{document} , $[eqn]$ , so that term can also be seen as a part of the regular paracontrolled remainder.

By assumption we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\sharp }\in \mathscr {L}_{T,p}^{\gamma ',\alpha +2\beta -1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\prime }\in (\mathscr {L}_{T,p}^{\gamma ',\alpha +\beta -1})^d$$\end{document} . This implies by (14) that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&f^{\sharp }\in \mathscr {M}^{0}_{\overline{T},T-(k-1)\overline{T}}\mathscr {C}^{\alpha +2\beta -1}_p\cap C^{1}_{\overline{T},T-(k-1)\overline{T}}\mathscr {C}^{2\beta -1}_p,\\ &f^{\prime }\in \mathscr {M}^{0}_{\overline{T},T-(k-1)\overline{T}}(\mathscr {C}^{\alpha +\beta -1}_p)^d\cap C^{1}_{\overline{T},T-(k-1)\overline{T}}(\mathscr {C}^{\beta -1}_p)^d \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2,\dots ,n$$\end{document} . If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{[T-\overline{T},T]}$$\end{document} denotes the solution on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-\overline{T},T]$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{[T-\overline{T},T]}_{T-\overline{T}}\in \mathscr {C}^{\alpha +\beta }_p$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{[T-\overline{T},T],\sharp }_{T-\overline{T}}\in \mathscr {C}^{2\alpha +2\beta -1}_p$$\end{document} . Thus, the solution on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-2\overline{T},T-\overline{T}]$$\end{document} follows

$$[eqn]$$

Because we can trivially bound,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{t\in [T-2\overline{T},T-\overline{T}]}\Vert P_{T-\overline{T}-t}u^{[T-2\overline{T},T-\overline{T}],\sharp }_{T-\overline{T}}\Vert _{\mathscr {C}^{2(\alpha +\beta )-1}_p}\lesssim \Vert u^{[T-2\overline{T},T-\overline{T}],\sharp }_{T-\overline{T}}\Vert _{\mathscr {C}^{2(\alpha +\beta )-1}_p}, \end{aligned}$$\end{document}

there is no blow-up for the solution on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-2\overline{T},T-\overline{T}]$$\end{document} at time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=T-\overline{T}$$\end{document} . Hence, the Banach fixed point argument for the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi ^{\overline{T},T-\overline{T}}$$\end{document} yields a solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{[T-2\overline{T},T-\overline{T}]}\in \mathscr {D}^{\hat{\gamma }}_{\overline{T},T-\overline{T}}$$\end{document} for any small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\gamma }>0$$\end{document} . By plugging the solution back in the fixed point map and using the interpolation estimates (cf. the arguments in the proof of Theorem 2 above and Corollary 2 below), we obtain that indeed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{[T-2\overline{T},T-\overline{T}]}\in \mathscr {D}^{0}_{\overline{T},T-\overline{T}}$$\end{document} . Proceeding iteratively, we thus obtain solutions

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u^{[T-k\overline{T},T-(k-1)\overline{T}]}\in \mathscr {D}^{0}_{\overline{T},T-(k-1)\overline{T}}\quad \text { for }\quad k=2,\dots ,n \end{aligned}$$\end{document}

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{[T-\overline{T},T]}\in \mathscr {D}^{\gamma }_{\overline{T},T-(k-1)\overline{T}}$$\end{document} . Then, the solution u, which is patched together on the subintervals ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{t}:=u^{[T-k\overline{T},T-(k-1)\overline{T}]}_{t}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in [T-k\overline{T},T-(k-1)\overline{T}]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1,\dots ,n$$\end{document} ), is indeed a fixed point of the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi =\phi ^{0,T}$$\end{document} considered on [0, T] and an element of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {D}_{T}^{\gamma }$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Corollary 2

By assumption, we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\prime }=0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\sharp }=u^{T}\in \mathscr {C}^{2(\alpha +\beta )-1}_{p}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\sharp },f^{\prime }$$\end{document} have no blow-up. By the assumption on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}$$\end{document} , it follows that $[eqn]$ due to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma '\in (0,1)$$\end{document} . Furthermore due to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\prime }=0$$\end{document} the paraproduct $[eqn]$ in (33) vanishes, which previously was the term that introduced a blow-up of at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma '$$\end{document} for the solution. Thus, we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{T-\cdot }u^{T}\in C_{T}\mathscr {C}^{2(\alpha +\beta )-1}$$\end{document} . Hence, the arguments from Theorem 3 yield a paracontrolled solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \mathscr {D}_{T}^{\gamma }$$\end{document} for any small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma >0$$\end{document} , i.p. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \mathscr {L}_{T,p}^{\gamma , \alpha +\beta }$$\end{document} . It remains to justify that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in D_{T}$$\end{document} . By the regular terminal condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}\in \mathscr {C}^{2(\alpha +\beta )-1}_p\subset \mathscr {C}^{\alpha +\beta }_p$$\end{document} and the interpolation estimate (29), we obtain that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{t\in [0,T]}\Vert u_{t}\Vert _{\mathscr {C}^{\alpha +\beta -\alpha \gamma }_p}&\lesssim \Vert u\Vert _{\mathscr {L}_{T,p}^{\gamma ,\alpha +\beta }}+\Vert u_{T}\Vert _{\mathscr {C}^{\alpha +\beta -\alpha \gamma }_p}\\ &\lesssim \Vert u\Vert _{\mathscr {L}_{T,p}^{\gamma ,\alpha +\beta }}+\Vert u_{T}\Vert _{\mathscr {C}^{\alpha +\beta }_p} \end{aligned}$$\end{document}

and since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{\sharp }_{T}=u_{T}\in \mathscr {C}^{2(\alpha +\beta )-1}_p$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{t\in [0,T]}\Vert u^{\sharp }_{t}\Vert _{\mathscr {C}^{2(\alpha +\beta )-1-\alpha \gamma }_p}&\lesssim \Vert u^{\sharp }\Vert _{\mathscr {L}_{T,p}^{\gamma ,2(\alpha +\beta )-1}}+\Vert u_{T}\Vert _{\mathscr {C}^{2(\alpha +\beta )-1-\gamma \alpha }_p}\\ &\lesssim \Vert u^{\sharp }\Vert _{\mathscr {L}_{T,p}^{\gamma ,2(\alpha +\beta )-1}}+\Vert u_{T}\Vert _{\mathscr {C}^{2(\alpha +\beta )-1}_p} \end{aligned}$$\end{document}

for any small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma >0$$\end{document} . If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is small enough, that is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,(3\beta +2\alpha -2)/\alpha )$$\end{document} , we can estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sup _{t\in [0,T]}\Vert \nabla u\cdot \mathscr {V}(t)\Vert _{\beta }\nonumber \\ &\lesssim \Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}(1+\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}})\Big (\sup _{t\in [0,T]}\Vert u_{t}\Vert _{\mathscr {C}^{\alpha +\beta -\alpha \gamma }_p}+\sup _{t\in [0,T]}\Vert u^{\sharp }_{t}\Vert _{\mathscr {C}^{2(\alpha +\beta )-1-\alpha \gamma }_p}\Big ). \end{aligned}$$\end{document}

Plugging now the solution u back in the contraction map using the fixed point, i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u=P_{T-\cdot }u^{T}+J^{T}(\nabla u\cdot \mathscr {V})$$\end{document} , and (38), we can use the Schauder estimates for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =\gamma '=0$$\end{document} , such that we obtain that indeed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \mathscr {L}_{T,p}^{0,\alpha +\beta }$$\end{document} . By the commutator estimate (21) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =\gamma '=0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \mathscr {L}_{T,p}^{0,\alpha +\beta }$$\end{document} , we then also obtain that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{\sharp }\in \mathscr {L}_{T,p}^{0,2(\alpha +\beta )-1}$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The next theorem proves the continuity of the solution map. The proof is similar to [2, Theorem 3.8], but adapted to the generalized setting for singular paracontrolled data. There are a few subtleties. First, the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {D}_{T}^{\gamma }(V,u^{T,\prime })$$\end{document} depends on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V, u^{T,\prime }$$\end{document} . Furthermore due to the blow-up \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma >0$$\end{document} , one cannot simply estimate the norm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {M}_{T}^{\gamma }\mathscr {C}^{\theta }$$\end{document} on the interval [0, T] by the sum of the respective blow-up norms on subintervals of [0, T]. In the case of regular terminal condition that splitting issue does not occur, but we aim for continuity of the solution map in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{T,p}^{0,\alpha +\beta }$$\end{document} . The latter we establish by first proving continuity of the map with values in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{T,p}^{\gamma ,\alpha +\beta }$$\end{document} for any small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma >0$$\end{document} and conclude from there together with the interpolation estimates.

Theorem 4

In the setting of Theorem 3, the solution map

$$[eqn]$$

is locally Lipschitz continuous, that is,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert u-v\Vert _{\mathscr {L}_{T,p}^{\gamma ',\alpha +\beta }}+\Vert u^{\sharp }-v^{\sharp }\Vert _{\mathscr {L}_{T,p}^{\gamma ,2(\alpha +\beta )-1}}\nonumber \\ &\quad \leqslant C[\Vert u^{T,\sharp }-v^{T,\sharp }\Vert _{\mathscr {C}^{(2-\gamma ')\alpha +2\beta -1}_p}+\Vert u^{T,\prime }-v^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^{d}}\nonumber \\ &\qquad +\Vert f^{\sharp }-g^{\sharp }\Vert _{\mathscr {L}^{\gamma ',\alpha +2\beta -1}_{T}}+\Vert f^{\prime }-g^{\prime }\Vert _{(\mathscr {L}^{\gamma ',\alpha +\beta -1}_{T})^{d}}+\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}] \end{aligned}$$\end{document}

for a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C=C(T,\Vert \mathscr {V}\Vert ,\Vert \mathscr {W}\Vert ,\Vert u^{T}\Vert ,\Vert v^{T}\Vert ,\Vert f\Vert ,\Vert g\Vert )>0$$\end{document} .

Furthermore, in the setting of Corollary 2, the solution map

$$[eqn]$$

is locally Lipschitz continuous allowing for an analogue bound (39) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma '=0$$\end{document} for the norms of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\sharp }-v^{T,\sharp },f^{\sharp }-g^{\sharp },f^{\prime }-g^{\prime }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\prime }=v^{T,\prime }=0$$\end{document} .

Proof

We first prove the continuity in the case of singular paracontrolled data.

Let u be the solution of the PDE for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}\in \mathscr {X}^{\beta ,\gamma '}$$\end{document} , $[eqn]$ and $[eqn]$ and v the solution corresponding to the data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {W}$$\end{document} , g and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v^{T}$$\end{document} . By the fixed point property we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (u,u')=(u,u')$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (v,v')=(v,v')$$\end{document} and thus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u'=\nabla u-f'$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v'=\nabla v-g'$$\end{document} . Hence, we can estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u'-v'\Vert _{(\mathscr {L}_{T,p}^{\gamma ',\alpha +\beta -1})^d}\lesssim \Vert u-v\Vert _{\mathscr {L}_{T,p}^{\gamma ',\alpha +\beta }}+\Vert f'-g'\Vert _{(\mathscr {L}_{T,p}^{\gamma ',\alpha +\beta -1})^d}. \end{aligned}$$\end{document}

We estimate the terms in (39) by itself times a factor less than 1, plus a term depending on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert f-g\Vert $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \mathscr {V}-\mathscr {W}\Vert $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert u^{T}-v^{T}\Vert $$\end{document} . Here we keep in mind that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \mathscr {D}^{\gamma }_{T}(V,u^{T,\prime })$$\end{document} , whereas \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v\in \mathscr {D}^{\gamma }_{T}(W,v^{T,\prime })$$\end{document} , but we explained the notation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert u - v\Vert _{\mathscr {D}^{\gamma }_{T}}$$\end{document} in Definition 3. For that purpose, we estimate the product using re-bracketing like \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ab-cd=a(b-d)+(a-c)d$$\end{document} and the estimate (36) for the product, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ''<\gamma '$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert \nabla u\cdot \mathscr {V}-\nabla v\cdot \mathscr {W}\Vert _{\mathscr {M}_{T}^{\gamma ''}\mathscr {C}^{\beta }_p} \nonumber \\ &\lesssim (1+\Vert \mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}})\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\Vert u-v\Vert _{\mathscr {D}^{\gamma }_{T}}+(1+\Vert \mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}})\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\Vert v\Vert _{\mathscr {D}^{\gamma }_{T}}\nonumber \\ &\qquad + \Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\Vert u\Vert _{\mathscr {D}^{\gamma }_{T}}\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\nonumber \\ &\qquad + \tilde{C}(\Vert \mathscr {V}\Vert ,\Vert \mathscr {W}\Vert ,\Vert u^{T,\prime }\Vert ,\Vert v^{T,\prime }\Vert )\big (\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}+\Vert u^{T,\prime }-v^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^{d}}\big ). \end{aligned}$$\end{document}

Since the solution u can be bounded in terms of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T},f,\mathscr {V}$$\end{document} by Gronwall’s inequality for locally finite measures using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ,\gamma '\in (0,1)$$\end{document} (cf. [21, Appendix, Theorem 5.1]), and similarly for v, we conclude that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert \nabla u\cdot \mathscr {V}-\nabla v\cdot \mathscr {W}\Vert _{\mathscr {M}_{T}^{\gamma ''}\mathscr {C}^{\beta }_p} \\ &\quad \lesssim \big ((\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}+\Vert v\Vert _{\mathscr {D}^{\gamma }_{T}})(1+\Vert \mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}})+\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\Vert u\Vert _{\mathscr {D}^{\gamma }_{T}}\big )\\ &\qquad \times \Big (\Vert u-v\Vert _{\mathscr {D}^{\gamma }_{T}}+\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\Big )\\ &\qquad +\tilde{C}\big (\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}+\Vert u^{T,\prime }-v^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^{d}}\big )\\ &\quad \lesssim C\Big (\Vert u-v\Vert _{\mathscr {D}^{\gamma }_{T}}+\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}+\Vert u^{T,\prime }-v^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^{d}}\Big ), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C=C(\Vert \mathscr {V}\Vert ,\Vert \mathscr {W}\Vert ,\Vert u^{T}\Vert ,\Vert v^{T}\Vert ,\Vert f\Vert ,\Vert g\Vert )$$\end{document} is a constant that depends on the norms of the input data on [0, T]. Therefore, we obtain by the fixed point and using the estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \phi (u,u^{\prime })\Vert _{\mathscr {L}_{T,p}^{\gamma ',\alpha +\beta }}$$\end{document} from the proof of Theorem 3 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ''<\gamma '$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert u-v\Vert _{\mathscr {L}_{T,p}^{\gamma ',\alpha +\beta }}\\ &\lesssim \Vert u^{T,\sharp }-v^{T,\sharp }\Vert _{\mathscr {C}^{(2-\gamma ')\alpha +2\beta -1}_p}+\Vert u^{T,\prime }-v^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^d}\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\nonumber \\ &\qquad +\Vert u^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^d}\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}+\Vert f^{\prime }\Vert _{(\mathscr {L}^{\gamma ',\alpha +\beta -1}_{T,p})^d}\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\nonumber \\ &\qquad +\Vert f^{\sharp }-g^{\sharp }\Vert _{\mathscr {L}^{\gamma ',\alpha +2\beta -1}_{T,p}}+\Vert f^{\prime }-g^{\prime }\Vert _{(\mathscr {L}^{\gamma ',\alpha +\beta -1}_{T,p})^d}\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\nonumber \\ &\qquad +T^{\gamma '-\gamma ''}\Vert \nabla u\cdot \mathscr {V}-\nabla v\cdot \mathscr {W}\Vert _{\mathscr {M}_{T}^{\gamma ''}\mathscr {C}^{\beta }_p}. \end{aligned}$$\end{document}

Moreover, using the fixed point and the estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \phi (u,u')^{\sharp }\Vert _{\mathscr {L}_{T,p}^{\gamma ,2(\alpha +\beta )-1}}$$\end{document} , we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert u^{\sharp }-v^{\sharp }\Vert _{\mathscr {L}_{T,p}^{\gamma ,2(\alpha +\beta )-1}}\nonumber \\&\lesssim \Vert u^{T,\sharp }-v^{T,\sharp }\Vert _{\mathscr {C}^{(2-\gamma ')\alpha +2\beta -1}_p}+\Vert u^{T,\prime }-v^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^d}\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\nonumber \\&\qquad +\Vert u^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^d}\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}+\Vert f^{\prime }\Vert _{(\mathscr {L}^{\gamma ',\alpha +\beta -1}_{T,p})^d}\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\nonumber \\&\qquad +\Vert f^{\sharp }-g^{\sharp }\Vert _{\mathscr {L}^{\gamma ',\alpha +2\beta -1}_{T,p}}+\Vert f^{\prime }-g^{\prime }\Vert _{(\mathscr {L}^{\gamma ',\alpha +\beta -1}_T)^d}\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\nonumber \\&\qquad +T^{\gamma -\gamma '}\Vert \nabla u\cdot \mathscr {V}-\nabla v\cdot \mathscr {W}\Vert _{\mathscr {M}_{T}^{\gamma '}\mathscr {C}^{\beta }_p}\nonumber \\&\qquad +T^{\gamma -\gamma '}\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\Vert u\Vert _{\mathscr {D}_{T}^{\gamma }} + T^{\gamma '-\gamma }\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\Vert u-v\Vert _{\mathscr {D}_{T}^{\gamma }}. \end{aligned}$$\end{document}

To shorten notation, let us abbreviate the term in (39) that we aim to estimate, in the following by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u-v\Vert _{\gamma ,\alpha +\beta }:=\Vert u-v\Vert _{\mathscr {L}_{T,p}^{\gamma ',\alpha +\beta }}+\Vert u^{\sharp }-v^{\sharp }\Vert _{\mathscr {L}_{T,p}^{\gamma ,2(\alpha +\beta )-1}}. \end{aligned}$$\end{document}

Then overall, using also (40), we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u-v\Vert _{\gamma ,\alpha +\beta }&\leqslant C[\Vert u^{T,\sharp }-v^{T,\sharp }\Vert _{\mathscr {C}^{(2-\gamma ')\alpha +2\beta -1}_p}+\Vert u^{T,\prime }-v^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^d}\nonumber \\&\qquad +\Vert f^{\sharp }-g^{\sharp }\Vert _{\mathscr {L}^{\gamma ',\alpha +2\beta -1}_{T,p}}\nonumber \\&\qquad +\Vert f^{\prime }-g^{\prime }\Vert _{(\mathscr {L}^{\gamma ',\alpha +\beta -1}_{T,p})^d}+\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}]\nonumber \\&\qquad +(T^{\gamma -\gamma '}\vee T^{\gamma '-\gamma ''})C\Vert u-v\Vert _{\gamma ,\alpha +\beta }, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C> 0$$\end{document} is again a (possibly different) constant depending on the norms of the input data. Assume for the moment that T is small enough so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(T^{\gamma -\gamma '}\vee T^{\gamma '-\gamma ''})C$$\end{document} times the implicit constant on the right-hand side is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<1$$\end{document} . Then we can take the last term to the other side and divide by a positive factor, obtaining

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u-v\Vert _{\gamma ,\alpha +\beta }&\leqslant C[\Vert u^{T,\sharp }-v^{T,\sharp }\Vert _{\mathscr {C}^{(2-\gamma ')\alpha +2\beta -1}_p}+\Vert u^{T,\prime }-v^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^d}\nonumber \\&\qquad +\Vert f^{\sharp }-g^{\sharp }\Vert _{\mathscr {L}^{\gamma ',\alpha +2\beta -1}_{T,p}}\nonumber \\&\qquad +\Vert f^{\prime }-g^{\prime }\Vert _{(\mathscr {L}^{\gamma ',\alpha +\beta -1}_{T,p})^d}+\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}], \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C=C(T,\Vert \mathscr {V}\Vert ,\Vert \mathscr {W}\Vert ,\Vert u^{T}\Vert ,\Vert v^{T}\Vert ,\Vert f\Vert ,\Vert g\Vert )>0$$\end{document} is a constant that depends on the norms of the input data. Thus, the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u^{T},f,\mathscr {V})\mapsto (u,u^{\sharp })$$\end{document} is locally Lipschitz continuous, which implies the claim.

If T is such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(T^{\gamma -\gamma '}\vee T^{\gamma '-\gamma ''})C$$\end{document} times the implicit constant is at least 1, then we want to apply the estimates above on the subintervals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-k\overline{T},T-(k-1)\overline{T}]$$\end{document} of length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}$$\end{document} is chosen, such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\overline{T}^{\gamma -\gamma '}\vee \overline{T}^{\gamma '-\gamma ''})C$$\end{document} times the implicit constant is strictly less than 1 and where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1,\dots ,n$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathbb {N}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T-n\overline{T}\leqslant 0$$\end{document} . To obtain the continuity in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {D}^{\gamma }_{T}$$\end{document} , we consider the solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{[T-k\overline{T},T-(k-1)\overline{T}]},v^{[T-k\overline{T},T-(k-1)\overline{T}]}$$\end{document} on the subintervals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-k\overline{T},T-(k-1)\overline{T}]$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1,\dots n$$\end{document} , where the terminal condition of the solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{[T-k\overline{T},T-(k-1)\overline{T}]}$$\end{document} is the initial value of the solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{[T-(k-1)\overline{T},T-(k-2)\overline{T}]}$$\end{document} (analogously for v), such that, patched together, we obtain the solutions u, v on [0, T]. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} to be chosen below. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2,\dots ,n$$\end{document} , we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{[T-k\overline{T},T-(k-1)\overline{T}]},v^{[T-k\overline{T},T-(k-1)\overline{T}]}\in \mathscr {D}_{T-(k-1)\overline{T}}^{0,\alpha +\beta }$$\end{document} (see the argument in the proof of Theorem 3), such that we can estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert u-v\Vert _{\mathscr {M}_{T}^{\gamma '}\mathscr {C}^{\alpha +\beta -\varepsilon \alpha }_p}\nonumber \\&\quad \leqslant T^{\gamma '}\Vert u-v\Vert _{\mathscr {M}^{0}_{T-\overline{T}}\mathscr {C}^{\alpha +\beta -\varepsilon \alpha }_p}+\Vert u-v\Vert _{\mathscr {M}_{\overline{T},T}^{\gamma '}\mathscr {C}^{\alpha +\beta }_p}\nonumber \\ &\quad \leqslant T^{\gamma '}\sum _{k=2}^{n}\Vert u-v\Vert _{\mathscr {M}^{0}_{\overline{T},T-(k-1)\overline{T}}\mathscr {C}^{\alpha +\beta -\varepsilon \alpha }_p}+\Vert u-v\Vert _{\mathscr {M}^{\gamma '}_{\overline{T},T}\mathscr {C}^{\alpha +\beta }_p}. \end{aligned}$$\end{document}

Furthermore, we can estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \in (0,\gamma ']$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u-v\Vert _{C_{T}^{1-\gamma '}\mathscr {C}^{\beta }_p}&\leqslant T^{\gamma '-\varepsilon }\Vert u-v\Vert _{C_{T-\overline{T}}^{1-\varepsilon }\mathscr {C}^{\beta }_p}+\Vert u-v\Vert _{C_{\overline{T},T}^{1-\gamma '}\mathscr {C}^{\beta }_p}\nonumber \\ &\leqslant T^{\gamma '-\varepsilon }\sum _{k=2}^{n}\Vert u-v\Vert _{\mathscr {L}_{\overline{T},T-(k-1)\overline{T},p}^{\varepsilon ,\beta +\alpha }}+\Vert u-v\Vert _{\mathscr {L}_{\overline{T},T,p}^{\gamma ',\beta +\alpha }}. \end{aligned}$$\end{document}

Subtracting the terminal condition for each of the terms with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2,\dots ,n$$\end{document} and applying the interpolation bound (29) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =\alpha +\beta $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\theta }=\varepsilon \alpha $$\end{document} yield for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2,\dots ,n$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert (u-u_{T-(k-1)\overline{T}})-(v-v_{T-(k-1)\overline{T}})\Vert _{\mathscr {M}^{0}_{\overline{T},T-(k-1)\overline{T}}\mathscr {C}^{\alpha +\beta -\varepsilon \alpha }_p}\nonumber \\ &\quad \leqslant \Vert (u-u_{T-(k-1)\overline{T}})-(v-v_{T-(k-1)\overline{T}})\Vert _{\mathscr {M}^{\varepsilon }_{\overline{T},T-(k-1)\overline{T}}\mathscr {C}^{\alpha +\beta }_p}. \end{aligned}$$\end{document}

Together with (44), (45) and (46), this then yields

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hspace{0.1em}&\hspace{-0.1em} \Vert u-v\Vert _{\mathscr {L}_{T,p}^{\gamma ',\alpha +\beta -\varepsilon \alpha }}\nonumber \\&\quad \lesssim T^{\gamma '}\sum _{k=2}^{n}\Big (\Vert (u-u_{T-(k-1)\overline{T}})-(v-v_{T-(k-1)\overline{T}})\Vert _{\mathscr {L}_{\overline{T},T-(k-1)\overline{T},p}^{0,\alpha +\beta -\varepsilon \alpha }}\end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\quad \quad +\Vert u_{T-(k-1)\overline{T}}-v_{T-(k-1)\overline{T}}\Vert _{\mathscr {C}^{\alpha +\beta }_p}\Big )\nonumber \\ &\quad \quad +\Vert u-v\Vert _{\mathscr {L}_{\overline{T},T,p}^{\gamma ',\alpha +\beta }}\nonumber \\&\quad \lesssim T^{\gamma '}\sum _{k=2}^{n}\Big (\Vert (u-u_{T-(k-1)\overline{T}})-(v-v_{T-(k-1)\overline{T}})\Vert _{\mathscr {L}_{\overline{T},T-(k-1)\overline{T},p}^{\varepsilon ,\alpha +\beta }}\end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\quad \quad +\Vert u_{T-(k-1)\overline{T}}-v_{T-(k-1)\overline{T}}\Vert _{\mathscr {C}^{\alpha +\beta }_p}\Big )\nonumber \\ &\quad \quad +\Vert u-v\Vert _{\mathscr {L}_{\overline{T},T,p}^{\gamma ',\alpha +\beta }}\nonumber \\ &\quad \lesssim T^{\gamma '}\sum _{k=2}^{n}\Vert u-v\Vert _{\mathscr {L}_{\overline{T},T-(k-1)\overline{T},p}^{\varepsilon ,\alpha +\beta }}+\Vert u-v\Vert _{\mathscr {L}_{\overline{T},T,p}^{\gamma ',\alpha +\beta }}, \end{aligned}$$\end{document}

where in the last estimate, we estimated the norm of the terminal conditions by the norm of the solutions in the previous iteration step.

Analogously, we can argue for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{\sharp }-v^{\sharp }$$\end{document} , obtaining

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert u^{\sharp }-v^{\sharp }\Vert _{\mathscr {L}_{T,p}^{\gamma ,2(\alpha +\beta )-1-\varepsilon \alpha }}\nonumber \\ &\quad \lesssim T^{\gamma }\sum _{k=2}^{n}\Vert u^{\sharp }-v^{\sharp }\Vert _{\mathscr {L}_{\overline{T},T-(k-1)\overline{T},p}^{\varepsilon ,2(\alpha +\beta )-1}}+\Vert u^{\sharp }-v^{\sharp }\Vert _{\mathscr {L}_{\overline{T},T,p}^{\gamma ,2(\alpha +\beta )-1}}. \end{aligned}$$\end{document}

Now, taking \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon :=(\gamma -\gamma ')\in (0,\gamma ')$$\end{document} , we can apply the above estimate (43) for each of the terms on the right-hand side of the inequalities (47) and (50). That is, for each of the terms for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2,\dots ,n$$\end{document} , we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hspace{0.5em}&\hspace{-0.5em} \Vert u-v\Vert _{\mathscr {L}_{\overline{T},T-(k-1)\overline{T},p}^{\varepsilon ,\alpha +\beta }}+\Vert u^{\sharp }-v^{\sharp }\Vert _{\mathscr {L}_{\overline{T},T-(k-1)\overline{T},p}^{\varepsilon ,2(\alpha +\beta )-1}}\\&\lesssim \frac{1}{{\displaystyle 1-\overline{T}^{\varepsilon }\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}(1+\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}})}}\\&\qquad \times \bigg [\Vert u^{T,\sharp }-v^{T,\sharp }\Vert _{\mathscr {C}^{(2-\gamma ')\alpha +2\beta -1}_p}+\Vert u^{T,\prime }-v^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^d}\nonumber \\ &\qquad +\Vert f^{\sharp }-g^{\sharp }\Vert _{\mathscr {L}^{\gamma ',\alpha +2\beta -1}_{T,p}}+\Vert f^{\prime }-g^{\prime }\Vert _{(\mathscr {L}^{\gamma ',\alpha +\beta -1}_{T,p})^d}+\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\bigg ]. \end{aligned}$$\end{document}

This uses that by the choice of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}^{\varepsilon }=\overline{T}^{\gamma -\gamma '}\leqslant \overline{T}^{\gamma -\gamma '}\vee \overline{T}^{\gamma '-\gamma ''}$$\end{document} and that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u,v\in \mathscr {D}_{T-(k-1)\overline{T}}^{0,\alpha +\beta }$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2,\dots ,n$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1$$\end{document} , we replace \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} , respectively, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma '$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{\sharp }-v^{\sharp }$$\end{document} , and obtain the estimate (43) on the subinterval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[T-\overline{T},T]$$\end{document} . Together, this then yields the following estimate on the whole interval [0, T] (with a possibly different constant C):

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert u-v\Vert _{\gamma ,\alpha +\beta -\varepsilon \alpha }\nonumber \\ &\quad \leqslant C\left[ \Vert u^{T,\sharp }-v^{T,\sharp }\Vert _{\mathscr {C}^{(2-\gamma ')\alpha +2\beta -1}_p}+\Vert u^{T,\prime }-v^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^d}\right. \nonumber \\ &\qquad \left. +\Vert f^{\sharp }-g^{\sharp }\Vert _{\mathscr {L}^{\gamma ',\alpha +2\beta -1}_{T,p}}+\Vert f^{\prime }-g^{\prime }\Vert _{(\mathscr {L}^{\gamma ',\alpha +\beta -1}_{T,p})^d}+\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\right] . \end{aligned}$$\end{document}

Plugging now \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u-v$$\end{document} back in the contraction map on [0, T], we can remove the loss \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \alpha $$\end{document} in regularity. That is, we can estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} small enough,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u-v\Vert _{\gamma ,\alpha +\beta }&\lesssim \Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}(1+\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}})\Vert u-v\Vert _{\gamma ,\alpha +\beta -\alpha \varepsilon } \\ &\quad + C[\Vert u^{T,\sharp }-v^{T,\sharp }\Vert _{\mathscr {C}^{(2-\gamma ')\alpha +2\beta -1}_p}+\Vert u^{T,\prime }-v^{T,\prime }\Vert _{(\mathscr {C}^{\alpha +\beta -1}_p)^d}\nonumber \\ &\quad +\Vert f^{\sharp }-g^{\sharp }\Vert _{\mathscr {L}^{\gamma ',\alpha +2\beta -1}_{T,p}}+\Vert f^{\prime }-g^{\prime }\Vert _{(\mathscr {L}^{\gamma ',\alpha +\beta -1}_{T,p})^d}+\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}]. \end{aligned}$$\end{document}

Thus the local Lipschitz continuity (39) on [0, T] follows.

In the setting of Corollary 2, we obtain from the above that the Lipschitz estimate (39) holds true with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma '=0$$\end{document} for the norms of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\sharp }-v^{T,\sharp },f^{\sharp }-g^{\sharp },f^{\prime }-g^{\prime }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T,\prime }=v^{T,\prime }=0$$\end{document} on the right-hand side and any small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma >0$$\end{document} on the left-hand side of the estimate. Similar as in the proof of Corollary 2, we can use the fixed point property and the estimate (38) for small enough \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma >0$$\end{document} , together with the Schauder estimates and the interpolation bound (29), to obtain that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hspace{0.1em}&\hspace{-0.1em} \Vert u-v\Vert _{\mathscr {L}_{T,p}^{0,\alpha +\beta }}+\Vert u^{\sharp }-v^{\sharp }\Vert _{\mathscr {L}_{T,p}^{0,2(\alpha +\beta )-1}}\\ &\lesssim \Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}(1+\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}})\Big (\Vert u-v\Vert _{\mathscr {L}_{T,p}^{0,\alpha +\beta -\gamma \alpha }}+\Vert u^{\sharp }-v^{\sharp }\Vert _{\mathscr {L}_{T,p}^{0,2(\alpha +\beta )-1-\gamma \alpha }}\Big )\\&\quad + C[\Vert u^{T,\sharp }-v^{T,\sharp }\Vert _{\mathscr {C}^{2\alpha +2\beta -1}_p}+\Vert f^{\sharp }-g^{\sharp }\Vert _{\mathscr {L}^{0,\alpha +2\beta -1}_{T,p}}\\&\quad +\Vert f^{\prime }-g^{\prime }\Vert _{(\mathscr {L}^{0,\alpha +\beta -1}_{T,p})^d}+\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}] \\&\lesssim \Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}(1+\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}})\\&\quad \times \Big (\Vert (u-v)-(u^{T,\sharp }-v^{T,\sharp })\Vert _{\mathscr {L}_{T,p}^{0,\alpha +\beta -\gamma \alpha }}+\Vert (u^{\sharp }-v^{\sharp })\\&\quad -(u^{T,\sharp }-v^{T,\sharp })\Vert _{\mathscr {L}_{T,p}^{0,2(\alpha +\beta )-1-\gamma \alpha }}\Big )\\&\quad + C[\Vert u^{T,\sharp }-v^{T,\sharp }\Vert _{\mathscr {C}^{2\alpha +2\beta -1}_p}+\Vert f^{\sharp }-g^{\sharp }\Vert _{\mathscr {L}^{0,\alpha +2\beta -1}_{T,p}}\\&\quad +\Vert f^{\prime }-g^{\prime }\Vert _{(\mathscr {L}^{0,\alpha +\beta -1}_{T,p})^d}+\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}] \\&\lesssim \Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}(1+\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}})\Big (\Vert u-v\Vert _{\mathscr {L}_{T,p}^{\gamma ,\alpha +\beta }}+\Vert u^{\sharp }-v^{\sharp }\Vert _{\mathscr {L}_{T,p}^{\gamma ,2(\alpha +\beta )-1}}\Big )\\&\quad + C[\Vert u^{T,\sharp }-v^{T,\sharp }\Vert _{\mathscr {C}^{2\alpha +2\beta -1}_p}+\Vert f^{\sharp }-g^{\sharp }\Vert _{\mathscr {L}^{0,\alpha +2\beta -1}_{T,p}}\\&\quad +\Vert f^{\prime }-g^{\prime }\Vert _{(\mathscr {L}^{0,\alpha +\beta -1}_{T,p})^d}+\Vert \mathscr {V}-\mathscr {W}\Vert _{\mathscr {X}^{\beta ,\gamma '}}]. \end{aligned}$$\end{document}

Notice that, to apply the interpolation bound (29) in the last estimate above, we subtracted the terminal condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{T}-v^{T}=u^{T,\sharp }-v^{T,\sharp }$$\end{document} , so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u-v)_{T}-(u^{T}-v^{T})=0$$\end{document} . The constant C above changes in each line. Thus together the Lipschitz continuity of the solution map with values in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}_{T,p}^{0,\alpha +\beta }\times \mathscr {L}_{T,p}^{0,2(\alpha +\beta )-1}$$\end{document} follows. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Next we consider solutions of the Kolmogorov PDE for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {G}^{\mathscr {V}}$$\end{document} (for fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}$$\end{document} ) on subintervals [0, r] of [0, T] for bounded sets of terminal conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(y^{r})_{r\in [0,T]}$$\end{document} and right-hand sides \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(f^{r})_{r\in [0,T]}$$\end{document} . In certain situations one is interested in a uniform bound on the norms of the solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u^{r})$$\end{document} on [0, r]. We prove the latter in the following corollary.

The solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{r}$$\end{document} on [0, r] has the following paracontrolled structure

$$[eqn]$$

with

$$[eqn]$$

for the commutators from the proof of Theorem 3.

Corollary 3

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T>0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}\in \mathscr {X}^{\beta ,\gamma '}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta ,\gamma '$$\end{document} as in Theorem 3. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (\gamma ',1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ''\in (0,\gamma ')$$\end{document} be as in the proof of Theorem 3. Let $[eqn]$ be a bounded sequence of singular paracontrolled terminal conditions, that is,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C_{y}:=\sup _{r\in [0,T]}[\Vert y^{r,\sharp }\Vert _{\mathscr {C}^{(2-\gamma ')\alpha +2\beta -1}_p}+ \Vert y^{r,\prime }\Vert _{\mathscr {C}^{\alpha +\beta -1}_p}]<\infty . \end{aligned}$$\end{document}

Let $[eqn]$ be a sequence of right-hand sides with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C_{f}:=\sup _{r\in [0,T]}[\Vert f^{r,\sharp }\Vert _{\mathscr {L}_{r,p}^{\gamma ',\alpha +2\beta -1}}+\Vert f^{r,\prime }\Vert _{\mathscr {L}_{r,p}^{\gamma ',\alpha +\beta -1}}]<\infty . \end{aligned}$$\end{document}

Let for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\in [0,T]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u^{r}_{t})_{t\in [0,r]}$$\end{document} be the solution of the backward Kolmogorov PDE for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {G}^{\mathscr {V}}$$\end{document} with terminal condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{r}_{r}=y^{r}$$\end{document} and right-hand side \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{r}$$\end{document} .

Then, the following uniform bound for the solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u^{r})$$\end{document} holds true

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sup _{r\in [0,T]}[\Vert u^{r,\sharp }\Vert _{\mathscr {L}_{r,p}^{\gamma ,2(\alpha +\beta )-1}}+\Vert u^{r}\Vert _{\mathscr {L}_{r,p}^{\gamma ',\alpha +\beta }}]\nonumber \\ &\quad \lesssim _{T} \lambda _{\overline{T},\mathscr {V}}^{-1}\bigg (\sup _{r\in [0,T]}[\Vert y^{r,\sharp }\Vert _{\mathscr {C}^{(2-\gamma ')\alpha +2\beta -1}_p}+\Vert f^{r,\sharp }\Vert _{\mathscr {L}_{r,p}^{\gamma ',\alpha +2\beta -1}}] \nonumber \\ &\quad \qquad +\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\sup _{r\in [0,T]}[\Vert y^{r,\prime }\Vert _{\mathscr {C}^{\alpha +\beta -1}_p}+\Vert f^{r,\prime }\Vert _{\mathscr {L}_{r,p}^{\gamma ',\alpha +\beta -1}}]\bigg ), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{\overline{T},\mathscr {V}}:=1-(\overline{T}^{\gamma -\gamma '}\vee \overline{T}^{ \gamma '-\gamma ''})\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}(1+\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}})>0$$\end{document} .

In particular, replacing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y^{r}$$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y^{r}_{1}-y^{r}_{2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{r}$$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{r}_{1}-f^{r}_{2}$$\end{document} with analogue bounds, a uniform Lipschitz bound for the solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{r}_{1}-u^{r}_{2}$$\end{document} follows.

In the setting of Corollary 2, the bound (53) holds true with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =\gamma '=0$$\end{document} , under the assumption, that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{f}+C_{y}<\infty $$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma '=0$$\end{document} .

Remark 12

In setting of Theorem 2 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} in the Young regime and considering bounded sets of terminal conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(y^{r})_{r}\subset \mathscr {C}^{(1-\gamma )\alpha +\beta }_p$$\end{document} and right-hand sides \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{r}\subset \mathscr {L}_{r,p}^{\gamma ,\beta }$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [0,1)$$\end{document} , an analogue uniform Lipschitz bound for the solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u^{r})$$\end{document} on [0, r] holds true. The proof is similar except much easier.

Proof

The proof follows from Theorem 4 replacing T by r and considering paracontrolled solutions on [0, r] in the sense of (52). Then, by (41) and (42) from the proof of Theorem 4 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {V}=\mathscr {W}$$\end{document} and splitting the interval [0, r] in subintervals of length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}$$\end{document} , we obtain for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\leqslant T$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert u^{r,\sharp }\Vert _{\mathscr {L}_{r,p}^{\gamma ,2(\alpha +\beta )-1}}+\Vert u^{r}\Vert _{\mathscr {L}_{r,p}^{\gamma ',\alpha +\beta }}\\ &\quad \lesssim _{T} C(r)\lambda _{\overline{T},\mathscr {V}}^{-1}\bigg (\sup _{r\in [0,T]}[\Vert y^{r,\sharp }\Vert _{\mathscr {C}^{(2-\gamma ')\alpha +2\beta -1}_p}+\Vert f^{r,\sharp }\Vert _{\mathscr {L}_{r,p}^{\gamma ',\alpha +2\beta -1}}] \nonumber \\ &\quad \qquad \qquad +\Vert \mathscr {V}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\sup _{r\in [0,T]}[\Vert y^{r,\prime }\Vert _{\mathscr {C}^{\alpha +\beta -1}_p}+\Vert f^{r,\prime }\Vert _{\mathscr {L}_{r,p}^{\gamma ',\alpha +\beta -1}}]\bigg ). \end{aligned}$$\end{document}

The dependence of the constant C(r) on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\leqslant T$$\end{document} is as follows: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(r)\lesssim \frac{r}{\overline{T}}\leqslant \frac{T}{\overline{T}}$$\end{document} . Notice that the choice of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{T}$$\end{document} only depends on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \mathscr {V}\Vert $$\end{document} , which is fixed here. Thus we obtain (53). As the solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{r}$$\end{document} depends linearly on the terminal condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y^{r}$$\end{document} and the right-hand side \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{r}$$\end{document} , the uniform Lipschitz bound follows. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Remark 13

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(V^{m})$$\end{document} be such that $[eqn]$ in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {X}^{\beta ,\gamma '}$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(f^{r})$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(y^{r})$$\end{document} be as in the corollary. Moreover, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(y^{r,m})$$\end{document} with $[eqn]$ be such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sup _{r\in [0,T]}\Vert y^{r,\sharp ,m}-y^{r,\sharp }\Vert _{\mathscr {C}_{p}^{(2-\gamma ')\alpha +2\beta -1}}\rightarrow 0$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\rightarrow \infty $$\end{document} . Analogously, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(f^{r,m})$$\end{document} with $[eqn]$ and convergence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(f^{r,\sharp ,m})_m$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{r}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{r,m}$$\end{document} be the solutions for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {G}^{\mathscr {V}}$$\end{document} with right-hand side \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{r}$$\end{document} and terminal conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y^{r}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y^{r,m}$$\end{document} , respectively. Then the proof of the corollary furthermore shows that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sup _{r\in [0,T]}[\Vert u^{r,\sharp }-u^{r,\sharp ,m}\Vert _{\mathscr {L}_{r,p}^{\gamma ,2(\alpha +\beta )-1}}+\Vert u^{r}-u^{r,m}\Vert _{\mathscr {L}_{r,p}^{\gamma ',\alpha +\beta }}]\nonumber \\ &\quad \lesssim _{T} \lambda _{\overline{T},\mathscr {V}}^{-1}\big (\sup _{r\in [0,T]}[\Vert y^{r,\sharp }-y^{r,\sharp ,m}\Vert _{\mathscr {C}^{(2-\gamma ')\alpha +2\beta -1}_p}+\Vert f^{r,\sharp }-f^{r,\sharp ,m}\Vert _{\mathscr {L}_{r,p}^{\gamma ',\alpha +2\beta -1}}] \nonumber \\ &\quad \qquad +\Vert \mathscr {V}-\mathscr {V}^{m}\Vert _{\mathscr {X}^{\beta ,\gamma '}}\sup _{r\in [0,T]}[\Vert y^{r,\prime }\Vert _{\mathscr {C}^{\alpha +\beta -1}_p}+\Vert f^{r,\prime }\Vert _{\mathscr {L}_{r,p}^{\gamma ',\alpha +\beta -1}}]\big )\\ &\quad \rightarrow 0, \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\rightarrow \infty $$\end{document} .