Bilinear embedding for divergence-form operators with complex coefficients on irregular domains

TL;DR

This paper extends bilinear inequalities for divergence-form operators with complex coefficients on irregular domains, establishing maximal regularity for associated parabolic problems without regularity assumptions on the domain boundary.

Contribution

It generalizes previous bilinear inequalities to irregular domains and proves maximal regularity for parabolic equations with complex coefficients under minimal boundary regularity.

Findings

Extended bilinear inequality to irregular domains

Proved maximal regularity for parabolic problems with complex coefficients

Established optimal exponent range for operator class

Abstract

Let $\Omega\subseteq \mathbb{R}^{d}$ be open and $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $\Omega$ with $L^{\infty}$ coefficients. Consider the divergence-form operator ${\mathscr L}^{A}=-{\rm div}(A\nabla)$ with mixed boundary conditions on $\Omega$. We extend the bilinear inequality that we proved in [16] in the special case when $\Omega=\mathbb{R}^{d}$. As a consequence, we obtain that the solution to the parabolic problem $u^{\prime}(t)+{\mathscr L}^{A}u(t)=f(t)$, $u(0)=0$, has maximal regularity in $L^{p}(\Omega)$, for all $p>1$ such that $A$ satisfies the $p$-ellipticity condition that we introduced in [16]. This range of exponents is optimal for the class of operators we consider. We do not impose any conditions on $\Omega$, in particular, we do not assume any regularity of $\partial\Omega$, nor the existence of a Sobolev embedding. The methods of [16] do not apply directly to the present case and a new argument is needed.

Equations

$$\mathrm{Re}⟨A(x)\xi ,\xi ⟩$$

$$\left|⟨A(x)\xi ,\sigma ⟩\right|$$

$${\text{gota}}_{A,\mathcal{V}}(u,v)={\int }_{\Omega }{⟨A\nabla u,\nabla v⟩}_{{\mathbb{C}}^d},\mathrm{D}({\text{gota}}_{A,\mathcal{V}})=\mathcal{V}.$$

$$\mathrm{D}\left({\mathcal{L}}_2^A\right):=\left\{u\in \mathcal{V}:\exists w\in L^2(\Omega ):{\text{gota}}_{A,\mathcal{V}}(u,v)={⟨w,v⟩}_{L^2(\Omega )}\forall v\in \mathcal{V}\right\}$$

$${\int }_{\Omega }{\mathcal{L}}_2^A(u)\overline{v}={\text{gota}}_{A,\mathcal{V}}(u,v),\forall u\in \mathrm{D}({\mathcal{L}}_2^A),\forall v\in \mathcal{V}.$$

$${\vartheta }_2^{\ast }:=\sup \left\{\arg ({\text{gota}}_{A,\mathcal{V}}(u,u)):u\in \mathcal{V}\right\}<\pi /2.$$

$${\mathbf{S}}_{{\vartheta }_2}=\{z\in \mathbb{C}\setminus \{0\}:|\arg (z)|<{\vartheta }_2\},$$

$$W_D^{1,2}(\Omega )={\overline{\{u_{{|}_{\Omega }}:u\in C_c^{\infty }({\mathbb{R}}^d\setminus D)\}}}^{W^{1,2}(\Omega )}.$$

$${\mathcal{I}}_p(\xi )=\xi +(1-2/p)\overline{\xi },\xi \in {\mathbb{C}}^d.$$

$${\Delta }_p(A):=\underset{x\in \Omega }{\mathrm{essinf}}\underset{|\xi |=1}{\min }\mathrm{Re}{⟨A(x)\xi ,{\mathcal{I}}_p\xi ⟩}_{{\mathbb{C}}^d}.$$

$${\Delta }_p(A)>0.$$

$$\mathrm{Re}{⟨A(x)\xi ,{\mathcal{I}}_p\xi ⟩}_{{\mathbb{C}}^d}⩾C|\xi {|}^2,\forall \xi \in {\mathbb{C}}^d.$$

$${\Delta }_p(A,B)$$

$$\lambda (A,B)$$

$$\Lambda (A,B)$$

$${\Delta }_p(A,B)>0⟹{\int }_0^{\infty }{\int }_{{\mathbb{R}}^d}\left|\nabla T_t^{A}f(x)\right|\left|\nabla T_t^{B}g(x)\right|\text{d}x\text{d}t⩽C{\Vert f\Vert }_p{\Vert g\Vert }_q,$$

$${\Delta }_p(A,B)>0⟹{\int }_0^{\infty }{\int }_{\Omega }\left|\nabla T_t^{A,\mathcal{V}}f(x)\right|\left|\nabla T_t^{B,{\mathcal{V}}^{\prime }}g(x)\right|\text{d}x\text{d}t⩽C{\Vert f\Vert }_p{\Vert g\Vert }_q,$$

$$\mathcal{V}\hookrightarrow L^q(\Omega ),$$

$$\mathcal{E}(t)={\int }_{\Omega }\mathcal{Q}(T_t^{A,\mathcal{V}}f,T_t^{B,{\mathcal{V}}^{\prime }}g)$$

$$\mathcal{Q}(\zeta ,\eta )=|\zeta {|}^p+|\eta {|}^q+\delta \{\begin{array}{ll}|\zeta {|}^2|\eta {|}^{2-q},& |\zeta {|}^p⩽|\eta {|}^q;\\ (2/p)|\zeta {|}^p+\left(2/q-1\right)|\eta {|}^q,& |\zeta {|}^p⩾|\eta {|}^q,\end{array}$$

$${\rm Y}=\{\eta =0\}\cup \{|\zeta {|}^p=|\eta {|}^q\}.$$

$$0⩽\mathcal{Q}(\zeta ,\eta )$$

$$|({\partial }_{\zeta }\mathcal{Q})(\zeta ,\eta )|$$

$$|({\partial }_{\eta }\mathcal{Q})(\zeta ,\eta )|$$

$$F_r(\gamma )=|\gamma {|}^r,\gamma \in {\mathbb{R}}^2,r>0.$$

$${\mathcal{V}}_d({\xi }_1+i{\xi }_2)=({\xi }_1,{\xi }_2),{\xi }_1,{\xi }_2\in {\mathbb{R}}^d.$$

$${\mathcal{W}}_{k,d}:\underset{k-\mathrm{times}}{\underbrace{{\mathbb{C}}^d\times \cdots \times {\mathbb{C}}^d}}⟶\underset{k-\mathrm{times}}{\underbrace{{\mathbb{R}}^{2d}\times \cdots \times {\mathbb{R}}^{2d}}},$$

$${\mathcal{W}}_{k,d}({\xi }^1,\dots ,{\xi }^k)=\left({\mathcal{V}}_d({\xi }^1),\dots ,{\mathcal{V}}_d({\xi }^k)\right),{\xi }^j\in {\mathbb{C}}^d,j=1,\dots ,k.$$

$$J=\left(\begin{array}[]{cc}0&-I_{\mathbb{R}^{d}}\\ I_{\mathbb{R}^{d}}&0\end{array}\right).$$

$${⟨\xi ,{\xi }^{\prime }⟩}_{{\mathbb{C}}^d}={⟨{\mathcal{V}}_d(\xi ),{\mathcal{V}}_d({\xi }^{\prime })⟩}_{{\mathbb{R}}^{2d}}+i{⟨{\mathcal{V}}_d(\xi ),J{\mathcal{V}}_d({\xi }^{\prime })⟩}_{{\mathbb{R}}^{2d}},\xi ,{\xi }^{\prime }\in {\mathbb{C}}^d.$$

$${\mathcal{M}}(A)={\mathcal{V}}_{d}A{\mathcal{V}}_{d}^{-1}=\left(\begin{array}[]{rr}{\rm Re}\,A&-{\rm Im}\,A\\ {\rm Im}\,A&{\rm Re}\,A\end{array}\right)\,.$$

$${⟨A\xi ,\sigma ⟩}_{{\mathbb{C}}^d}={⟨\mathcal{M}(A){\mathcal{V}}_d(\xi ),{\mathcal{V}}_d(\sigma )⟩}_{{\mathbb{R}}^{2d}}+i{⟨J^t\mathcal{M}(A){\mathcal{V}}_d(\xi ),{\mathcal{V}}_d(\sigma )⟩}_{{\mathbb{R}}^{2d}},$$

$$\mathrm{Re}{⟨A\xi ,\sigma ⟩}_{{\mathbb{C}}^d}={⟨\mathcal{M}(A){\mathcal{V}}_d(\xi ),{\mathcal{V}}_d(\sigma )⟩}_{{\mathbb{R}}^{2d}}.$$

Full text

1. Introduction

Let $\Omega\subseteq\mathbb{R}^{d}$ be a nonempty open set. Denote by ${\mathcal{A}}(\Omega)$ the class of all complex uniformly strictly elliptic $d\times d$ matrix-valued functions on $\Omega$ with $L^{\infty}$ coefficients (in short, elliptic matrices). That is to say ${\mathcal{A}}(\Omega)$ is the class of all measurable $A:\Omega\rightarrow{\mathbb{C}}^{d\times d}$ for which there exist $\lambda=\lambda(A),\Lambda=\Lambda(A)>0$ such that for almost all $x\in\Omega$ we have

[TABLE]

Suppose that $A\in{\mathcal{A}}(\Omega)$. Fix a closed subspace ${\mathscr{V}}$ of $W^{1,2}(\Omega)$ containing $W^{1,2}_{0}(\Omega)$. Denote by ${\mathscr{L}}^{A}_{2}={\mathscr{L}}^{A,{\mathscr{V}}}_{2}$ the unbounded operator on $L^{2}(\Omega)$ associated with the densely defined, accretive, continuous and closed sesquilinear form

[TABLE]

Namely,

[TABLE]

and

[TABLE]

Ellipticity of $A$ implies that the form ${\text{\got a}}_{A,{\mathscr{V}}}$ is sectorial in the sense of Kato:

[TABLE]

Therefore, the associated operator is the negative generator of a strongly continuous semigroup $(T^{A,{\mathscr{V}}}_{t})_{t>0}$ on $L^{2}(\Omega)$ which is analytic and contractive in the cone

[TABLE]

where $\vartheta_{2}=\pi/2-\vartheta^{*}_{2}$. We also have ${\mathscr{L}}^{A^{*}}_{2}=({\mathscr{L}}^{A})^{*}_{2}$, so $T^{A^{*}}_{t}=(T^{A}_{t})^{*}$ for all $t>0$. For details and proofs, see [Kat, Chapter VI], [AuTc] and [O, Chapters I and IV].

Given a closed set $D\subseteq\partial\Omega$ we define

[TABLE]

1.1. Mixed boundary conditions

We shall always assume that ${\mathscr{V}}$ is one of the following closed subspaces of $W^{1,2}(\Omega)$:

1. (i)

${\mathscr{V}}=W^{1,2}(\Omega)$ corresponding to Neumann boundary conditions for ${\mathscr{L}}^{A}$, or 2. (ii)

${\mathscr{V}}=W^{1,2}_{D}(\Omega)$ corresponding to Dirichlet conditions in $D$ and Neumann conditions in $\partial\Omega\setminus D$ for ${\mathscr{L}}^{A}$.

The latter case includes Dirichlet boundary conditions ($D=\partial\Omega$) and good-Neumann boundary conditions ($D=\emptyset$); see [O, Section 4.1].

We notice that the very same boundary conditions have been recently considered, for example, in [Egert, Egert2018, Elst2019] but under stronger assumptions on $\Omega$.

In the special case of pure Neumann boundary conditions ${\mathscr{V}}=W^{1,2}(\Omega)$ we denote the semigroup generated by $-{\rm div}(A\nabla)$, $A\in{\mathcal{A}}(\Omega)$, simply by $(T^{A}_{t})_{t>0}$.

1.2. The $p$-ellipticity condition

We summarize the following notion, which we introduced in [CD-DivForm].

For every $p\in[1,+\infty]$ consider the $\mathbb{R}$-linear operator

[TABLE]

Given $A\in{\mathcal{A}}(\Omega)$ and $p\in[1,+\infty]$, we introduce the number

[TABLE]

We say that $A$ is $p$-elliptic if

[TABLE]

By definition, $A$ is $p$-elliptic if and only if there exists $C=C(A,p)>0$ such that for a.e. $x\in\Omega$,

[TABLE]

Clearly, $\Delta_{2}(A)>0$ is a reformulation of the ellipticity condition (1). It follows from the definition that a bounded matrix function $A$ is real and elliptic if and only if it is $p$-elliptic for all $p>1$. For further properties of the function $p\mapsto\Delta_{p}(A)$ we refer the reader to [CD-DivForm] and Lemma 4.

Dindoš and Pipher in [Dindos-Pipher] and [Dindos-Pipher2, Dindos-Pipher3] showed that the key condition (5) also bears deep connections with the regularity theory of elliptic PDE. They found the sharp condition which permits proving reverse Hölder inequalities for weak solutions to ${\rm div}(A\nabla u)=0$ with complex $A$. It turns out that this condition is precisely a reformulation of $p$-ellipticity (5).

Recently, Egert [Egert2018] and, independently, ter Elst, Haller-Dintelmann, Rehberg and Tolksdorf [Elst2019] used $p$-ellipticity and its properties for studying semigroup extrapolation and parabolic maximal regularity for divergence form operators with mixed boundary conditions on domains $\Omega$ that satisfy certain geometric assumptions.

A condition similar to (5), namely $\Delta_{p}(A)\geqslant 0$, was formulated in a different manner by Cialdea and Maz’ya in [CiaMaz, (2.25)]. It was a result of their study of a condition on forms known as $L^{p}$-dissipativity. We arrived in [CD-DivForm] at the $p$-ellipticity, and thus also at $\Delta_{p}(A)\geqslant 0$, from another direction (bilinear embeddings and generalized convexity of power functions) further developing and extending the methods from [CD-mult] and [CD-OU]; see [CD-DivForm, Remark 5.9].

1.3. Semigroup estimates and bilinear embedding on $\mathbb{R}^{d}$

In [CD-DivForm, Theorem 1.3] we used a theorem of Nittka [Nittka, Theorem 4.1] and showed that the condition $\Delta_{p}(A)\geqslant 0$ implies contractivity in $L^{p}(\Omega)$ of the semigroup generated by $-{\rm div}\,(A\nabla)$ with Dirichlet boundary conditions in $\Omega$. This improved an earlier result of Cialdea and Maz’ya [CiaMaz]. A straightforward modification of the proof shows that the implication $(a)\Rightarrow(b)$ in [CD-DivForm, Theorem 1.3] still holds true if we replace Dirichlet boundary conditions with Neumann boundary conditions. One can also consider mixed boundary conditions, see [Egert2018]. So we have the following result.

Proposition 1**.**

Suppose that $\Omega\subseteq\mathbb{R}^{d}$ is open. Let ${\mathscr{V}}$ be one of the subspaces of Section 1.1. Let $A\in{\mathcal{A}}(\Omega)$ and $p>1$ be such that $\Delta_{p}(A)\geqslant 0$. Then $(T^{A,{\mathscr{V}}}_{t})_{t>0}$ extends to a strongly continuous semigroup of contractions on $L^{p}(\Omega)$.

One of the main points of [CD-DivForm] was the connection between $p$-ellipticity and bilinear embeddings associated with divergence-form operators with complex coefficients. More specifically, given $A,B\in{\mathcal{A}}(\Omega)$ define

[TABLE]

In the special case $\Omega=\mathbb{R}^{d}$ we proved in [CD-DivForm, Theorem 1.1] that there exists $C>0$, depending only on $p$, $\Delta_{p}(A,B)$, $\lambda(A,B)$ and $\Lambda(A,B)$, such that

[TABLE]

for all $f,g\in(L^{p}\cap L^{q})(\mathbb{R}^{d})$.

The $p$-ellipticity condition appeared while we were studying in [CD-DivForm] the validity of the right-hand side of (7) and is related to the notion of generalised convexity, or convexity with respect to matrices, that we previously studied in some specific cases in [CD-mult] and [CD-OU] and that we shall discuss in Section 2.

1.4. Bilinear embedding on domains

While Proposition 1 and [CD-DivForm, Theorem 1.3] hold true for all open $\Omega\subseteq\mathbb{R}^{d}$, in [CD-DivForm] we were able to prove the bilinear estimate (7) only in the special case $\Omega=\mathbb{R}^{d}$. One of the targets of the present paper is to extend (7) to every open set $\Omega\subset\mathbb{R}^{d}$. In Section 6 we shall prove the following result.

Theorem 2**.**

Suppose that $\Omega\subset\mathbb{R}^{d}$ is an open set. Let ${\mathscr{V}}$ and ${\mathscr{V}}^{\prime}$ be two closed subspaces of $W^{1,2}(\Omega)$ of the type described in Section 1.1. Let $p>1$, $q=p/(p-1)$ and $A,B\in{\mathcal{A}}(\Omega)$. Then there exists $C>0$, depending only on $p$, $\Delta_{p}(A,B)$, $\lambda(A,B)$ and $\Lambda(A,B)$, such that

[TABLE]

for all $f,g\in(L^{p}\cap L^{q})(\Omega)$.

The method we used in [CD-DivForm] for proving (7) does not apply to arbitrary open $\Omega\subset\mathbb{R}^{d}$, even for pure Neumann boundary conditions ${\mathcal{V}}={\mathcal{V}}^{\prime}=W^{1,2}(\Omega)$. Indeed, we proved (7) by means of a regularisation argument [CD-DivForm, Section 6] which reduces the proof to the case of smooth $A,B\in{\mathcal{A}}(\mathbb{R}^{d})$ with bounded derivatives. The reduction procedure was used for justifying the integration by parts behind the formula [CD-DivForm, (3.3)]; see [CD-DivForm, Section 4.2]. In $\mathbb{R}^{d}$ the advantage of working with smooth coefficients with bounded derivatives is that by results of Auscher, McIntosh and Tchamitchian [AMT, Theorem 4.15] and Auscher [AuscherNuc, Theorem 4.8], in this case, $T^{A}_{t}$ and $T^{B}_{t}$ are bounded in $L^{\infty}(\mathbb{R}^{d})$, for all $t>0$; this was the property we used for proving [CD-DivForm, Theorem 1.1].

In Section 6.2 we shall simplify the proof of [CD-DivForm, Theorem 1.1] by means of a new argument based on the aforementioned regularisation trick and elliptic regularity [AgDoNi] for smooth coefficient operators. The key fact here is that if $A$ is sufficiently regular then the domain of ${\mathscr{L}}^{A}$ in $L^{r}(\mathbb{R}^{d})$ coincides with $W^{2,r}(\mathbb{R}^{d})$, $1<r<+\infty$. This makes it possible to work with the operator core $C^{\infty}_{c}(\mathbb{R}^{d})$, so that all the integrations by parts can be easily justified.

For divergence-form operators ${\mathscr{L}}^{A}$ on $\Omega\subset\mathbb{R}^{d}$ with, say, Neumann boundary conditions ${\mathscr{V}}=W^{1,2}(\Omega)$ the situation is different. On one hand the domain of the Neumann Laplacian ${\mathscr{L}}^{I}$ in $L^{p}(\Omega)$ is unknown and, in general, it is not included in $W^{2,p}(\Omega)$ [Gris, Costabel] or even in $W^{1,p}(\Omega)$ [JeKe]. One the other hand, extrapolation of $T^{A}_{t}$ on $L^{\infty}(\Omega)$ is not expected, even for complex constant $A$ (see Section A.5) and it is not clear if there exists an operator core of bounded functions for ${\mathscr{L}}^{A}_{p}$, $1<p<\infty$. This makes the regularisation procedure used in [CD-DivForm] and Section 6.2 useless for the proof of (8), and forces us to modify the Bellman-function-heat-flow method we used in [CD-DivForm]; see Section 3. This is the main technical novelty of the present paper.

1.5. Maximal regularity and functional calculus on domains

By means of the elementary properties of the function $(p,A)\mapsto\Delta_{p}(A)$ (see Lemma 4), a general result of Cowling, Doust, McIntosh, and Yagi [CDMY, Theorem 4.6 and Example 4.8] and the Dore-Venni theorem [DoreVenni, PrussSohr] we deduce from Theorem 2 applied with $B=A^{*}$ and ${\mathscr{V}}^{\prime}={\mathscr{V}}$ the following result; see Section 7.

Theorem 3**.**

Suppose that $\Omega\subseteq\mathbb{R}^{d}$ is an open set. Let ${\mathscr{V}}$ be one of the subspaces of Section 1.1. Let $A\in{\mathcal{A}}(\Omega)$ and $p>1$. Suppose that $A$ is $p$-elliptic (that is $\Delta_{p}(A)>0$). Then the negative generator ${\mathscr{L}}^{A}_{p}$ of $(T^{A,{\mathscr{V}}}_{t})_{t>0}$ on $L^{p}(\Omega)$ admits a bounded holomorphic functional calculus of angle $<\pi/2$. As a consequence, ${\mathscr{L}}^{A}_{p}$ has parabolic maximal regularity.

The novelty of Theorem 3 lies in the fact that we are able to prove parabolic maximal regularity for some $p\neq 2$ without assuming any regularity of the boundary $\partial\Omega$, nor the existence of a Sobolev embedding

[TABLE]

for some $q>2$. Hence our results complement those of [Egert2018, Elst2019]; see Section A.

Two results of Kunstmann [Kunstmann3, Example 2.4 and Remark 3.2] and [kunstmann2] show that the range of $p$’s in Theorem 3 is optimal even for the class $\{{\mathscr{L}}^{A}:A\in{\mathcal{A}}(\Omega)\cap C^{\infty}(\Omega),\ {\mathscr{V}}=W^{1,2}(\Omega),|\Omega|<+\infty\}$. This means that, given $d\geqslant 2$, there exist $\Omega_{0}\subset\mathbb{R}^{d}$ of finite measure and $A_{0}\in{\mathcal{A}}(\Omega)$ with smooth coefficients such that ${\mathscr{L}}^{A_{0}}$ subject to Neumann boundary conditions in $\Omega_{0}$ has parabolic maximal regularity in $L^{p}(\Omega_{0})$ if and only if $\Delta_{p}(A_{0})>0$; see Section A.4.

2. Heat-flow monotonicity and generalised convexity

For proving the bilinear inequality (8) we use a variant of the Bellman-function-heat-flow method originally introduced by Petermichl and Volberg in [PV] and Nazarov and Volberg in [NV], and extended in [DV-Sch, Dv-kato, CD-Riesz]. Here we further refine the “complex-time” version of this method that we developed in [CD-mult, CD-OU, CD-DivForm]. This new refinement addresses a major technical issue (see Section 3).

2.1. The Bellman function of Nazarov and Treil

In the context of the present paper this method consists of studying the monotonicity of the flow111In the definition of ${\mathcal{E}}$ we implicitly identify $T^{A,{\mathscr{V}}}_{t}f$ and $T^{B,{\mathscr{V}}^{\prime}}_{t}g$ with their real counterparts; see Section 2.4 for a more accurate statement.

[TABLE]

associated with a particular explicit Bellman function ${\mathcal{Q}}$ invented by Nazarov and Treil [NT] in 1995. Here we use a simplified variant introduced in [Dv-kato] which comprises only two variables:

[TABLE]

where $p>2$, $q=p/(p-1)$, $\zeta,\eta\in\mathbb{R}^{2}$ and $\delta>0$ is a positive parameter that will be fixed later. Recall from [CD-DivForm] that ${\mathcal{Q}}\in C^{1}(\mathbb{R}^{4})\cap C^{2}(\mathbb{R}^{4}\setminus\Upsilon)$, where

[TABLE]

For $(\zeta,\eta)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$ we have

[TABLE]

where $\partial_{\zeta}=\left(\partial_{\zeta_{1}}-i\partial_{\zeta_{2}}\right)/2$ and $\partial_{\eta}=\left(\partial_{\eta_{1}}-i\partial_{\eta_{2}}\right)/2$.

The construction of the original Nazarov–Treil function was one of the earliest examples of the so-called Bellman function technique, which was introduced in harmonic analysis shortly beforehand by Nazarov, Treil and Volberg [NTV]. The name “Bellman function” stems from the stochastic optimal control, see [NTV1] for details. The same paper [NTV1] explains the connection between the Nazarov–Treil–Volberg approach and the earlier work of Burkholder on martingale inequalities; see [Bu1] and [Bu2, Bu4]. For an in-depth treatise on recent advances in martingale inequalities the reader is referred to [Os]. If interested in the genesis of Bellman functions and the overview of the method, the reader is also referred to [V, NT, W]. Recent applications of Bellman-heat-flow methods include [DomPe, PSW, CD-Riesz, CD-mult, CD-OU, CD-DivForm, MaSp, Dah, Wrobel1, BDFS, DV, DV-Sch, DraVol].

A formal passage of the time derivative under the integral sign in (9) and a more delicate formal integration by parts (see the discussion in Section 1.4) suggest that the monotonicity of ${\mathcal{E}}$ is related to the convexity properties of ${\mathcal{Q}}$; see Section 2.4. Indeed, it naturally leads to a new notion of convexity called generalised convexity with respect to the matrices $(A,B)$; in short $(A,B)$-convexity [CD-mult, CD-OU, CD-DivForm].

Owing to the tensor structure of ${\mathcal{Q}}$, the generalised convexity of ${\mathcal{Q}}$ is related to that of its elementary building blocks (see [CD-DivForm] and Section 2.5): the power functions

[TABLE]

It turns out that $F_{p}$ is $A$-convex if and only if $\Delta_{p}(A)\geqslant 0$, and ${\mathcal{Q}}$ is strictly $(A,B)$-convex provided that $\Delta_{p}(A,B)>0$; see [CD-DivForm] and Theorem 6.

We now formalise the notion of generalised convexity.

2.2. Real form of complex operators

We explicitly identify ${\mathbb{C}}^{d}$ with $\mathbb{R}^{2d}$ as follows. For each $d\in{\mathbb{N}}_{+}$ consider the operator ${\mathcal{V}}_{d}:{\mathbb{C}}^{d}\rightarrow\mathbb{R}^{d}\times\mathbb{R}^{d}$, defined by

[TABLE]

Let $k,d\in{\mathbb{N}}_{+}$. We define another identification operator

[TABLE]

by the rule

[TABLE]

Denote by $J$ the standard symplectic operator on $\mathbb{R}^{2d}$ given by

[TABLE]

The operator $J$ is associated to the standard complex structure on $\mathbb{R}^{d}\times\mathbb{R}^{d}$. Namely $J$ is the real form of the multiplication by $i$: ${\mathcal{V}}_{d}(i\xi)=J{\mathcal{V}}_{d}(\xi)$ and

[TABLE]

If $A\in{\mathbb{C}}^{d\times d}$ we shall frequently use its real form:

[TABLE]

Observe that ${\mathcal{M}}(A^{*})={\mathcal{M}}(A)^{T}$ and ${\mathcal{M}}(AB)={\mathcal{M}}(A){\mathcal{M}}(B)$. For $\xi,\sigma\in{\mathbb{C}}^{d}$ we have

[TABLE]

In particular,

[TABLE]

2.3. Convexity with respect to complex matrices

Let $k\in{\mathbb{N}}_{+}$ and $V\subseteq\mathbb{R}^{2k}$ a open subset. Suppose that $\Phi:V\rightarrow\mathbb{R}$ is of class $C^{2}$. Denote by $(D^{2}\Phi)(\omega)$ the Hessian matrix of $\Phi$ at the point $\omega\in V$. Let $d\in{\mathbb{N}}_{+}$ and $A_{1},\dots,A_{k}\in{\mathbb{C}}^{d\times d}$. Denote by ${\mathcal{M}}(A^{*}_{1})\oplus\cdots\oplus{\mathcal{M}}(A^{*}_{k})$ the $2kd\times 2kd$ block diagonal real matrix with the $2d\times 2d$ blocks ${\mathcal{M}}(A^{*}_{1}),\dots,{\mathcal{M}}(A^{*}_{k})$ along the main diagonal. For each $\omega\in V$, we define the new matrix

[TABLE]

We call $H^{(A_{1},\dots,A_{k})}_{\Phi}(\omega)$ the generalised Hessian of $\Phi$ at the point $\omega$ with respect to the complex matrices $A_{1},\dots,A_{k}$. We say that $\Phi$ is $(A_{1},\dots,A_{k})$-convex in $V$ if the quadratic form associated with $H^{(A_{1},\dots,A_{k})}_{\Phi}(\omega)$ is nonnegative at every $\omega\in V$. We shall often say that $\Phi$ is $(A_{1},\dots,A_{k})$-convex in a single point $\omega\in V$, if the condition above holds for that particular $\omega$. The same for $(A_{1},\dots,A_{k})$-convexity in a subset of $V$.

In accordance with [CD-DivForm], we introduce a special notation for denoting the quadratic form associated with the generalised Hessians. Given $A_{1},\dots,A_{k}\in{\mathbb{C}}^{d\times d}$ and $\Phi:V\subseteq\mathbb{R}^{2k}\rightarrow\mathbb{R}$, we define

[TABLE]

We maintain the same notation when instead of matrices we consider matrix-valued functions $A_{1},\dots,A_{k}\in L^{\infty}(\Omega;{\mathbb{C}}^{d\times d})$; in this case however we require that all the conditions are satisfied for a.e. $x\in\Omega$.

2.4. Heat-flow monotonicity

The main reason for introducing the notion of convexity with respect to complex matrices (generalised convexity) is its link with the monotonicity of certain functionals associated with semigroups [CD-mult, CD-OU, CD-DivForm]. In what follows we explain this link at a formal level. In the applications, the justification of the formal passages is part of the problem (see the discussion in Section 3), and it will not be addressed here.

Let $\Omega\subseteq\mathbb{R}^{d}$, $A_{1},\dots,A_{k}\in{\mathcal{A}}(\Omega)$ and ${\mathscr{V}}_{1},\dots,{\mathscr{V}}_{k}$ of the type described in Section 1.1. Let $\Phi:\mathbb{R}^{2k}\rightarrow\mathbb{R}_{+}$ be of class, say, $C^{2}$. Define $\Phi_{{\mathcal{W}}}=\Phi\circ{\mathcal{W}}_{k,1}$. Given $f_{1},\dots,f_{k}\in L^{2}(\Omega)$, define the function

[TABLE]

• a)

Suppose that we can differentiate and interchange derivative and integral. Then a calculation (see [CD-DivForm]) shows that

[TABLE]

where $\zeta^{j}=(\zeta^{j}_{1},\zeta^{j}_{2})\in\mathbb{R}^{k}\times\mathbb{R}^{k}$ and $\partial_{\zeta^{j}}=\left(\partial_{\zeta^{j}_{1}}-i\partial_{\zeta^{j}_{2}}\right)/2$.

• b)

Suppose that each $(\partial_{\zeta^{j}}\Phi)_{{\mathcal{W}}}(T^{A_{1}}_{t}f_{1},\dots,T^{A_{k}}_{t}f_{k})$ belongs to the form domain ${\mathscr{V}}_{j}$. Then we can integrate by parts in the sense of (3) on the right-hand side of (13) and, by means of another calculation (see [CD-DivForm]), we get

[TABLE]

It follows that if $\Phi$ is $(A_{1},\dots,A_{k})$-convex on $\mathbb{R}^{2k}$ then the function ${\mathcal{E}}$ is nonincreasing on $(0,+\infty)$.

When $\Phi$ is strictly $(A_{1},\dots,A_{k})$-convex and satisfies a suitable size estimate, this formal method can be used for proving bilinear inequalities in the spirit of [CD-DivForm, CD-mult, CD-OU].

2.5. Generalised convexity of power functions and the Bellman function of Nazarov and Treil

Let $\Omega\subseteq\mathbb{R}^{d}$ an open subset, $A\in{\mathcal{A}}(\Omega)$ and $p>1$. Recall the definition of the number $\Delta_{p}(A)$ in (4).

The following facts will be used in this paper. They were proven in [CD-DivForm].

Lemma 4**.**

[CD-DivForm]*

Let $A\in{\mathcal{A}}(\Omega)$, $p>1$ and $q=p/(p-1)$. Then*

1. (i)

$\Delta_{p}(A)=\Delta_{q}(A)$** 2. (ii)

$\Delta_{p}(A)\geqslant 0$* if and only if $\Delta_{p}(A^{*})\geqslant 0$. The same holds for strict inequalities.* 3. (iii)

The function $p\mapsto\Delta_{p}(A)$ is Lipschitz continuous and nonincreasing on $[2,\infty)$. 4. (iv)

The function $\varphi\mapsto\Delta_{p}(e^{i\varphi}A)$ is Lipschitz continuous in the interval $(-\pi/2,\pi/2)$. 5. (v)

For a.e. $x\in\Omega$ and every $\zeta\in\mathbb{R}^{2}\setminus\{0\}$ and $\xi\in{\mathbb{C}}^{d}$ we have

[TABLE]

In particular,

[TABLE]

for all $\zeta\in\mathbb{R}^{2}\setminus\{0\}$. 6. (vi)

${\rm Im}\,A(x)=0$* for a.e. $x\in\Omega$ if and only if $\Delta_{p}(A)\geqslant 0$ for all $p>1$.*

Remark 5**.**

In principle, Proposition 1 can be deduced by combining Lemma 4 (v) with the heat-flow method of Section 2.4 applied with $\Phi=F_{p}$. However this is just a formal argument and can not be directly used. This problem can be fixed by using a result of Nittka [Nittka, Theorem 4.1]; see [CD-DivForm, Theorem 1.3] and the beginning of Section 1.3.

Recall the notation (6). The next theorem establishes a link between generalized convexity of power functions $|\zeta|^{p}$ and $|\eta|^{q}$ and strict generalised convexity of the Bellman function ${\mathcal{Q}}={\mathcal{Q}}_{p,\delta}$ defined in (10).

Theorem 6** ([CD-DivForm, Theorem 5.2]).**

Suppose that $p\geqslant 2$ and $A,B\in{\mathcal{A}}(\Omega)$ satisfy $\Delta_{p}(A,B)>0$. Then there exists $\delta=\delta(\Delta_{p}(A,B),\lambda(A,B),\Lambda(A,B))\in(0,1)$ such that the function ${\mathcal{Q}}$ is strictly $(A,B)$-convex in $\mathbb{R}^{4}\setminus\Upsilon$. More specifically, for almost every $x\in\Omega$ we have

[TABLE]

for any $\omega\in\mathbb{R}^{4}\setminus\Upsilon$ and $X,Y\in\mathbb{R}^{2d}$.

3. Strategy for proving the bilinear embedding (Theorem 2)

Let $p>1$ and $A,B\in{\mathcal{A}}(\Omega)$ be such that $\Delta_{p}(A,B)>0$. To simplify the exposition, in this section we only consider pure Neumann boundary conditions ${\mathscr{V}}={\mathscr{V}}^{\prime}=W^{1,2}(\Omega)$ for ${\mathscr{L}}^{A}$ and ${\mathscr{L}}^{B}$. Ignore for one moment that the Bellman function ${\mathcal{Q}}$ defined in (10) is not globally $C^{2}$ (this can be easily fixed by means of convolution with smooth approximation of identity; see Section 5).

We would like to use the heat-flow method of Section 2.4 applied with $k=2$ and $\Phi={\mathcal{Q}}$ to deduce Theorem 2 from the strict $(A,B)$-convexity of ${\mathcal{Q}}$ (see Theorem 6), Proposition 1 and the first size estimate in (11). This was our approach in [CD-DivForm]. The major difficulty here is that it is not clear whether $(\partial_{\zeta}Q)(T^{A}_{t}f,T^{B}_{t}g)$ and $(\partial_{\eta}Q)(T^{A}_{t}f,T^{B}_{t}g)$ belong to the form domain $W^{1,2}(\Omega)$ whenever $f,g\in(L^{p}\cap L^{q})(\Omega)$, so the hypothesis of Section 2.4 b) may not be satisfied and we cannot justify the integration by parts (3) on the right-hand side of (13). As we remarked in the introduction, in the special case when $\Omega=\mathbb{R}^{d}$ we overcame this difficulty by using a regularisation argument that we learnt from [AuTc, Section 1.2]; see [CD-DivForm, Theorem 1.1, Section 4.2 and Appendix]. Since we do not see how to modify the regularisation trick in the case of general open subsets $\Omega\subset\mathbb{R}^{d}$, for proving Theorem 2 we instead modify the Bellman-heat-flow argument used in [CD-DivForm]; see Section 6.

Our idea is to approximate the Nazarov-Treil Bellman function ${\mathcal{Q}}$ with a sequence $({\mathcal{R}}_{n,\nu})_{n\in{\mathbb{N}}_{+}}$, $\nu>0$, of smooth $(A,B)$-convex functions having first order partial derivatives of linear growth and bounded second order derivatives (see Theorem 16), in a such a way that, for $\Phi={\mathcal{R}}_{n,\nu}$, the integration by parts in (13) is justified. Then we deduce Theorem 2 by a limiting argument. The construction of the sequence $({\mathcal{R}}_{n,\nu})_{n\in{\mathbb{N}}_{+}}$, although based on elementary arguments, requires some effort, because of the rigidity of the $(A,B)$-convexity.

We now shortly describe the main steps in the construction of the sequence $({\mathcal{R}}_{n,\nu})_{n\in{\mathbb{N}}_{+}}$. The technical details are postponed to Section 4. Denote by $\star$ the convolution in $\mathbb{R}^{4}$.

1. (a)

Since we need to use the chain-rule for the composition with vector-valued Sobolev functions [Leonichain, AmbMaschain], it is convenient to replace ${\mathcal{Q}}$ with its regularised version ${\mathcal{Q}}\star\varphi_{\nu}$, where $\{\varphi_{\nu}\}_{\nu\in(0,1)}$ is a smooth compactly supported approximation of the identity in $\mathbb{R}^{4}$; see Section 5. 2. (b)

The size of the first- and second-order derivatives of ${\mathcal{Q}}\star\varphi_{\nu}$ (see the estimates of Lemma 14) does not justify the integration by parts as in Section 2.4 with $\Phi={\mathcal{Q}}\star\varphi_{\nu}$. So we cut ${\mathcal{Q}}\star\varphi_{\nu}$ by means of a sequence of smooth mollifiers $\{\psi_{n}\}_{n\in{\mathbb{N}}_{+}}$ such that $\psi_{n}\geqslant 0$ is supported, say, in $B_{\mathbb{R}^{4}}(0,4n)$ and $\psi_{n}=1$ in $B_{\mathbb{R}^{4}}(0,3n)$. 3. (c)

The function $\psi_{n}\cdot({\mathcal{Q}}\star\varphi_{\nu})$ is not $(A,B)$-convex in the region $\{\omega\in\mathbb{R}^{4}:3n\leqslant|\omega|\leqslant 4n\}$. To fix this problem, we add another regular function ${\mathcal{P}}_{n,\nu}$ which is globally $(A,B)$-convex and strictly $(A,B)$-convex in the annulus, in order that

[TABLE]

becomes $(A,B)$-convex in all $\mathbb{R}^{4}$; see Section 5.

In the construction of ${\mathcal{P}}_{n,\nu}$ we need to bear in mind that:

• •

For each $n$ the function ${\mathcal{R}}_{n,\nu}$ must have partial derivatives with linear growth and bounded second order derivatives (needed for the integration by parts as in Section 2.4 b) with $\Phi={\mathcal{R}}_{n,\nu}$);

• •

$D^{2}{\mathcal{R}}_{n,\nu}$ must converge pointwise to $D^{2}({\mathcal{Q}}\star\varphi_{\nu})$ (needed for applying the strict $(A,B)$-convexity of ${\mathcal{Q}}\star\varphi_{\nu}$; see Proposition 13);

• •

$D{\mathcal{R}}_{n,\nu}$ must converge pointwise to $D({\mathcal{Q}}\star\varphi_{\nu})$ and satisfy

[TABLE]

uniformly in $n\in{\mathbb{N}}_{+}$ (needed for applying Lebesgue convergence theorem and passing to the limit as $n$ goes to $\infty$ in the right-hand side of (13) with $\Phi={\mathcal{R}}_{n,\nu}$).

We shall define

[TABLE]

for a suitable sequence $\{{\mathcal{P}}_{n}\}_{n\in{\mathbb{N}}_{+}}$ and constant $C(\nu)>0$; see Sections 4 and 5.

Since in the ball $\{\omega\in\mathbb{R}^{4}:|\omega|\leqslant 5n\}$ we have the estimate

[TABLE]

(see (34)), a natural choice (see, for example, [Yan]) for the sequence $\{{\mathcal{P}}_{n}\}_{n\in{\mathbb{N}}_{+}}$ would be ${\mathcal{P}}_{n}(\omega):=P_{n}(\omega):=p_{n}(|\omega|)$, where

[TABLE]

Unfortunately, Proposition B.1 applied with $\Gamma(\zeta)=P_{n}(\zeta,0)$ or $\Gamma(\zeta)=P_{n}(0,\zeta)$ for $\zeta\in\mathbb{R}^{2}$, shows that if either ${\rm Im}\,A\neq 0$ or ${\rm Im}\,B\neq 0$, the function $P_{n}$ is not $(A,B)$-convex in $\mathbb{R}^{4}\setminus\{|\omega|=n\}$.

By Lemma 4 (v), under the assumption $\Delta_{p}(A,B)>0$, the sum of the $2$-variable power functions

[TABLE]

is $(A,B)$-convex in $\mathbb{R}^{4}\setminus\{0\}$ and by Lemma 4 (iii) the range of $p$-ellipticity is open. So in (14) one can try to take ${\mathcal{P}}_{n}$ of the form

[TABLE]

where

[TABLE]

and $\varepsilon>0$ is such that $\Delta_{p+\varepsilon}(A,B)>0$. However, this is not enough to compensate for the lack of $(A,B)$-convexity of $\psi_{n}\cdot({\mathcal{Q}}\star\varphi_{\nu})$ (see point (c)) in regions of the form $\{\omega\in\mathbb{R}^{4}:3n\leqslant|\omega|\leqslant 4n\}\cap S_{\kappa}$, $\kappa>1$, where

[TABLE]

In light of the previous considerations, one can try to define ${\mathcal{P}}_{n}$ by means of suitable truncated $4$-variable power functions of the form

[TABLE]

where $f_{n}$ is given by (15). It turns out that even this is not the right sequence, since, in general, the condition $\Delta_{p}(A,B)>0$ does not imply that the $4$-variable power function

[TABLE]

is $(A,B)$-convex in $\mathbb{R}^{4}\setminus\{0\}$.

Example 7**.**

Consider the case when $A=aI_{{\mathbb{C}}^{d}}$ and $B=bI_{{\mathbb{C}}^{d}}$, $a,b>0$. Then by Lemma 4 (vi) we have $\Delta_{p}(A,B)>0$ for all $p>1$. By Lemma 8, on the other hand, $F_{p}$ is $(A,B)$-convex in $\mathbb{R}^{4}\setminus\{0\}$ if and only if $|1-2/p|\leqslant 2\sqrt{ab}/(a+b)$. The same result also follows from [Bakry3, Proof of Théorème 7 and p. 19].

We show in Proposition 10 that the $4$-variable power function $F_{p}$ is $(A,B)$-convex in a subregion $S_{\kappa}$ depending on $A$, $B$ and $p$.

We also show in Proposition 11 that in the complementary region, $\mathbb{R}^{4}\setminus S_{k}$, a suitable multiple of the sum of the $2$-variable power functions $G_{p}$ compensates for the lack of $(A,B)$-convexity of $F_{p}$ in $\mathbb{R}^{4}\setminus S_{k}$.

Finally, we end up with the right sequence $\{{\mathcal{P}}_{n}\}_{n\in{\mathbb{N}}_{+}}$:

[TABLE]

for suitable $\varepsilon>0$ and $K>0$ depending on $p$, $A$ and $B$, and $f_{n}$ as in (15).

4. The sequence $\{{\mathcal{P}}_{n}\}_{n\in{\mathbb{N}}_{+}}$

The aim of this section is to provide all the details in the construction of the sequence $\{{\mathcal{P}}_{n}\}_{n\in{\mathbb{N}}_{+}}$ roughly described in Section 3 and prove some of its properties. For the reader’s convenience we also recollect here some notation from Sections 1 and 2.

4.1. Power functions in higher dimensions

Let $p>1$ and $l\in{\mathbb{N}}_{+}$. Define $F_{p}:(\mathbb{R}^{2})^{l}\rightarrow\mathbb{R}_{+}$ by

[TABLE]

We remark that while the power functions defined above are different for different values of the dimension $l$, we will use the same symbol “$F_{p}$” to denote all of them.

For $\omega_{1},\dots,\omega_{l}\in\mathbb{R}^{2}$, $X_{1},\dots,X_{l}\in\mathbb{R}^{d}\times\mathbb{R}^{d}$ and $A_{1},\dots,A_{l}\in{\mathcal{A}}(\Omega)$, we set

[TABLE]

We also define

[TABLE]

Lemma 8**.**

Let $p>1$, $d,l\in{\mathbb{N}}_{+}$, $A_{1},\dots,A_{l}\in{\mathcal{A}}(\Omega)$, $\omega\in(\mathbb{R}^{2})^{l}$ and $X\in(\mathbb{R}^{d}\times\mathbb{R}^{d})^{l}$. Then, for $\omega\neq 0$,

[TABLE]

In case when $|\omega|=1$ we have, for

[TABLE]

the following formulæ :

1. (I)

[TABLE] 2. (II)

[TABLE]

Proof.

A rapid calculation shows that

[TABLE]

and (16) trivially follows from definitions.

Now assume that $|\omega|=1$. From (17) we get

[TABLE]

In order to calculate the $\Sigma_{1}$, write the summands as

[TABLE]

By applying (12) on the first term above and (17) and Lemma 4 (v) on the second, we get

[TABLE]

In order to calculate $\Sigma_{2}$, first write $\vartheta_{j}:=\arg\omega_{j}$, $j=1,\dots,l$. A calculation shows that

[TABLE]

Therefore,

[TABLE]

Observe that the last factor in the second term on the right-hand side is the real form of the complex conjugation in ${\mathbb{C}}^{d}$. Consequently, from the identity

[TABLE]

we get

[TABLE]

By using (19) with $k=j$ and (12) we conclude that

[TABLE]

Hence

[TABLE]

The identity (II) now follows by combining (18) with (20).

In order to prove (I), we write the diagonal terms in (II) as

[TABLE]

and use the identity

[TABLE]

We note that in the special case $l=1$, Lemma 8 is consistent with Lemma 4 (v).

Corollary 9**.**

Let $p\geqslant 2$, $d,l\in{\mathbb{N}}_{+}$ and $A_{1},\dots,A_{l}\in{\mathcal{A}}(\Omega)$. Then for every $\omega\in(\mathbb{R}^{2})^{l}$ with $|\omega|=1$ and for every $X\in(\mathbb{R}^{d}\times\mathbb{R}^{d})^{l}$ we have

[TABLE]

Proof.

For $j=1,\dots,l$ set

[TABLE]

From (II) we get

[TABLE]

Since $\Delta_{2}(B)\geqslant\Delta_{p}(B)$ for every $p>1$ and every matrix $B$, we may continue as

[TABLE]

Since $p\geqslant 2$, the corollary is proved. ∎

4.2. Generalised convexity of the $4$-variable power function

For each $\kappa>0$ we consider the subregion of $\mathbb{R}^{4}$ given by

[TABLE]

Note that for $\kappa\in(0,1]$ we have $S_{\kappa}=\mathbb{R}^{4}$. Also, when $\kappa>1$ and $\omega\in\mathbb{R}^{4}\setminus S_{\kappa}$, we have

[TABLE]

If $p\geqslant 2$ and $A,B\in{\mathcal{A}}(\Omega)$ are $p$-elliptic, then we define the constant

[TABLE]

Proposition 10**.**

Let $p>2$. Suppose that $A,B\in{\mathcal{A}}(\Omega)$ satisfy $\Delta_{p}(A,B)>0$. Let $\kappa\geqslant\kappa_{p}$. Then $F_{p}$ is $(A,B)$-convex in the region $S_{\kappa}$, that is,

[TABLE]

for all $\omega\in S_{\kappa}$ and for all $X,Y\in\mathbb{R}^{d}\times\mathbb{R}^{d}$.

Proof.

The proposition follows from (16) and Corollary 9, applied with $l=2$, and the definition of $\kappa_{p}$. ∎

4.3. Modified $4$-variable power function

We now perturb the $4$-variable power function $F_{p}$ in order to get a function $(A,B)$-convex in all of $\mathbb{R}^{4}$. Let $p>2$ and $A,B\in{\mathcal{A}}(\Omega)$ such that $\Delta_{p}(A,B)>0$. Define $\kappa_{p}$ by (22) and set

[TABLE]

Consider the function

[TABLE]

Proposition 11**.**

Suppose that $\Delta_{p}(A,B)>0$. Then $P_{p}$ is $(A,B)$-convex in $\mathbb{R}^{4}$.

Proof.

If $\kappa_{p}\leqslant 1$, then $S_{\kappa_{p}}=\mathbb{R}^{4}$ and $P_{p}=F_{p}$ on $\mathbb{R}^{4}$. Hence the proposition in this case follows from Proposition 10.

Suppose now that $\kappa_{p}>1$. Let $\omega=(\zeta,\eta)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$ and $X,Y\in\mathbb{R}^{d}\times\mathbb{R}^{d}$. We have

[TABLE]

Since $\Delta_{p}(A,B)>0$, Proposition 10 and Lemma 4 (v) imply that $P_{p}$ is $(A,B)$-convex in the region $S_{\kappa_{p}}$.

If $\omega\in\mathbb{R}^{4}\backslash{S_{\kappa_{p}}}$, we separately estimate the two terms in the right-hand side of (25). Since $\Delta_{p}(A,B)>0$, Lemma 8 and Lemma 4 (v) give

[TABLE]

while Lemma 4 (v) and (21) give

[TABLE]

In order to finish the proof, combine (25), (26) and (27). ∎

4.4. Definition of ${\mathcal{P}}_{n}$

Fix $p>2$ and $A,B\in{\mathcal{A}}(\Omega)$ such that $\Delta_{p}(A,B)>0$. By Lemma 4 (iii) there exists $\varepsilon=\varepsilon(p,A,B)>0$ such that $\Delta_{p+\varepsilon}(A,B)>0$. For this particular $\varepsilon>0$ and all $n\in{\mathbb{N}}_{+}$ define $f_{n}$ by (15). For every $l\in{\mathbb{N}}_{+}$ define ${\mathcal{F}}_{n}:(\mathbb{R}^{2})^{l}\rightarrow\mathbb{R}_{+}$ by

[TABLE]

Let $\kappa_{p+\varepsilon}$ and $K_{p+\varepsilon}$ be the two constants given by (22) and (23). We define

[TABLE]

For any $n\in{\mathbb{N}}_{+}$, consider the set $\Theta_{n}\subset\mathbb{R}^{4}$ defined by

[TABLE]

Proposition 12**.**

1. (i)

${\mathcal{P}}_{n}\in C^{1}(\mathbb{R}^{4})\cap C^{2}(\mathbb{R}^{4}\setminus\Theta_{n})$* for all $n\in{\mathbb{N}}_{+}$. Moreover,*

[TABLE]

as $n\rightarrow\infty$. 2. (ii)

${\mathcal{P}}_{n}$* is $(A,B)$-convex in $\mathbb{R}^{4}\setminus\Theta_{n}$, for all $n\in{\mathbb{N}}_{+}$. Moreover, for all $n\in{\mathbb{N}}_{+}$ and all $\omega\in\mathbb{R}^{4}\setminus\Theta_{n}$ with $|\omega|>n$, we have*

[TABLE] 3. (iii)

There exists $C>0$ that does not depend on $n$ such that

[TABLE] 4. (iv)

For every $n\in{\mathbb{N}}_{+}$ there exists $C(n)>0$ such that

[TABLE] 5. (v)

$D^{2}{\mathcal{P}}_{n}\in L^{\infty}(\mathbb{R}^{4}\setminus\Theta_{n};\mathbb{R}^{4\times 4})$, for all $n\in{\mathbb{N}}_{+}$.

Proof.

Item (i) is an immediate consequence of the definition of ${\mathcal{P}}_{n}$.

We now prove item (ii). Let $\omega=(\zeta,\eta)\in(\mathbb{R}^{2}\times\mathbb{R}^{2})\setminus\Theta_{n}$. Suppose first that $|\omega|<n$; then $|\zeta|<n$ and $|\eta|<n$. Hence, in this case, ${\mathcal{F}}_{n}(\zeta)=n^{-\varepsilon}F_{p+\varepsilon}(\zeta)$, ${\mathcal{F}}_{n}(\eta)=n^{-\varepsilon}F_{p+\varepsilon}(\eta)$ and ${\mathcal{F}}_{n}(\omega)=n^{-\varepsilon}F_{p+\varepsilon}(\omega)$. Therefore ${\mathcal{P}}_{n}(\omega)=n^{-\varepsilon}P_{p+\varepsilon}(\omega)$ for all $\omega\in\mathbb{R}^{4}\setminus\Theta_{n}$ with $|\omega|<n$ and the $(A,B)$-convexity follows from Proposition 11.

Suppose now that $|\omega|>n$. Then,

[TABLE]

Therefore,

[TABLE]

where $a=(p+\varepsilon)n^{p-2}$. Since

[TABLE]

and $\Delta_{p+\varepsilon}(A,B)>0$, we deduce from Lemma 4 that

[TABLE]

for all $\sigma_{1},\sigma_{2}\in{\mathbb{C}}^{d}$ and all $\omega\in\mathbb{R}^{4}\setminus\Theta_{n}$ with $|\omega|>n$. This finishes the proof of item (ii). Items (iii), (iv) and (v) easily follow from definitions. ∎

5. The sequence $\{{\mathcal{R}}_{n,\nu}\}_{n\in{\mathbb{N}}_{+}}$

Let $p>2$ and $q=p/(p-1)$. Fix $A,B\in{\mathcal{A}}(\Omega)$ with $\Delta_{p}(A,B)>0$. Let ${\mathcal{Q}}={\mathcal{Q}}_{\delta}$ denote the Nazarov-Treil Bellman function introduced in (10) with $\delta>0$ chosen so that Theorem 6 holds true.

Fix a radial function $\varphi\in C_{c}^{\infty}(\mathbb{R}^{4})$ such that $0\leqslant\varphi\leqslant 1$, ${\rm supp}\,\varphi\subset B_{\mathbb{R}^{4}}(0,1)$ and $\int\varphi=1$. Also, fix a radial function $\psi\in C^{\infty}_{c}(\mathbb{R}^{4})$ such that $\psi\geqslant 0$, $\psi=1$ on $\{|\omega|\leqslant 3\}$ and $\psi=0$ on $\{|\omega|>4\}$. For $\nu\in(0,1]$ and $n\in{\mathbb{N}}_{+}$ define $\varphi_{\nu}(\omega)=\nu^{-4}\varphi(\omega/\nu)$ and $\psi_{n}(\omega)=\psi(\omega/n)$.

Notation

Let $\{{\mathcal{P}}_{n}\}_{\in{\mathbb{N}}_{+}}$ be the sequence of Section 4.4. For every $n\in{\mathbb{N}}_{+}$ and all $\nu\in(0,1]$, define

[TABLE]

where $C_{1}=C_{1}(p,A,B,\psi)>0$ is a constant not depending on $\nu$ which will be fixed later.

5.1. Estimates for ${\mathcal{Q}}\star\varphi_{\nu}$

Next result was proven in [CD-DivForm, Corollary 5.5].

Proposition 13**.**

Suppose that $p\geqslant 2$ and $A,B\in{\mathcal{A}}(\Omega)$ satisfy $\Delta_{p}(A,B)>0$. Then ${\mathcal{Q}}\star\varphi_{\nu}$ is $(A,B)$-convex in $\mathbb{R}^{4}$. More specifically, for almost every $x\in\Omega$ we have

[TABLE]

for any $\omega\in\mathbb{R}^{4}$ and $X,Y\in\mathbb{R}^{2d}$.

We shall need estimates of the first- and second-order partial derivatives of $Q*\varphi_{\nu}$. As a consequence of (11) we have (recall that $\delta$ is fixed):

[TABLE]

for all $\zeta,\eta\in\mathbb{R}^{2}$ and $\nu\in(0,1)$, see [CD-Riesz, Theorem 4]. Also, a calculation shows that

[TABLE]

for all $(\zeta,\eta)\in\mathbb{R}^{4}\setminus\Upsilon$, where $\Upsilon$ is defined on page 9.

Lemma 14**.**

There exists $C=C(p,\delta)>0$ such that

1. (i)

${\displaystyle\hskip 5.0pt|({\mathcal{Q}}\star\varphi_{\nu})(\omega)|\leqslant C\left(|\omega|^{p}+|\omega|^{q}+1\right);}$** 2. (ii)

${\displaystyle\hskip 5.0pt\left|{D({\mathcal{Q}}\star\varphi_{\nu})(\omega)}\right|\leqslant C\left(|\omega|^{p-1}+|\omega|^{q-1}\right);}$** 3. (iii)

${\displaystyle\left|{D^{2}({\mathcal{Q}}\star\varphi_{\nu})(\omega)}\right|\leqslant C\nu^{q-2}\left(|\zeta|^{p-2}+|\eta|^{2-q}+1\right)}$**

for all $\nu\in(0,1]$ and $\omega=(\zeta,\eta)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$.

Proof.

Item (i) directly follows from the first estimate in (29). Item (iii) follows from (30) and the properties of convolution. Let us only treat in detail the convolution with the term with the negative exponent, $|\eta|^{q-2}$. We have

[TABLE]

Now we prove (ii). Let $j\in\{1,2,3,4\}$. Since ${\mathcal{Q}}$ and $\varphi_{\nu}$ are even functions in each of the variables $\zeta_{1},\zeta_{2},\eta_{1},\eta_{2}$, function ${\mathcal{Q}}\star\varphi_{\nu}$ also has this property, so

[TABLE]

Hence, by item (iii) and the mean value theorem, if $|\omega|<\nu\leqslant 1$ we get

[TABLE]

By the second and third estimate in (29), there exists $C>0$ not depending on $\nu\in(0,1)$ and such that

[TABLE]

5.2. Estimates for ${\mathcal{P}}_{n}\star\varphi_{\nu}$

Since ${\mathcal{P}}_{n}\in C^{1}(\mathbb{R}^{4})$ and its second-order partial derivatives exist on $\mathbb{R}^{4}\setminus\Theta_{n}$ and extend to a locally integrable function on $\mathbb{R}^{4}$, we have

[TABLE]

Proposition 15**.**

Let $\nu\in(0,1)$.

1. (i)

The functions $D({\mathcal{P}}_{n}\star\varphi_{\nu})$ and $D^{2}({\mathcal{P}}_{n}\star\varphi_{\nu})$ converge pointwise to [math] in $\mathbb{R}^{4}$ as $n\rightarrow\infty$. 2. (ii)

The function ${\mathcal{P}}_{n}\star\varphi_{\nu}$ is $(A,B)$-convex in $\mathbb{R}^{4}$. Moreover, for all $n\in{\mathbb{N}}_{+}$, $X,Y\in\mathbb{R}^{2d}$ and all $\omega$ with $|\omega|>2n$,

[TABLE] 3. (iii)

There exists $C>0$ that does not depend on $n$ and $\nu$ such that

[TABLE] 4. (iv)

For every $n\in{\mathbb{N}}_{+}$ there exists $C(n)>0$ (that does not depend on $\nu$) such that

[TABLE] 5. (v)

$D^{2}({\mathcal{P}}_{n}\star\varphi_{\nu})\in L^{\infty}(\mathbb{R}^{4};\mathbb{R}^{4\times 4})$* and ${\left\|{D^{2}({\mathcal{P}}_{n}\star\varphi_{\nu})}\right\|_{\infty}}\leqslant C(n)$ independently of $\nu$.*

Proof.

Item (i) follows by combining (32), Proposition 12 (i) and (iii) with the dominated convergence theorem. Item (v) follows from (32) and Proposition 12 (v).

By (32) we have

[TABLE]

for all $x\in\Omega$, $\omega\in\mathbb{R}^{4}$ and $X,Y\in\mathbb{R}^{2d}$. Since we assumed that $|\omega|>2n$ and since the support of the integrand is contained in $B_{\mathbb{R}^{4}}(0,\nu)$, we have $|\omega-\omega^{\prime}|>2n-\nu>n$, therefore we may estimate the integrand by means of Proposition 12 (ii) almost everywhere on $B_{\mathbb{R}^{4}}(0,\nu)$ and thus prove item (ii).

Let us address item (iii). We proceed much as in the proof of Lemma 14 (ii). First consider $|\omega|\leqslant 1$. The function ${\mathcal{P}}_{n}\star\varphi_{\nu}$ is smooth and even in $\zeta_{1},\zeta_{2},\eta_{1},\eta_{2}$, so

[TABLE]

Hence, the second identity in (32), the second estimate of Proposition 12 (iii) and the mean value theorem imply

[TABLE]

Now take $|\omega|>1$. From the first identity in (32) and the first estimate of Proposition 12 (iii) we get

[TABLE]

Thus we proved (iii).

Finally, item (iv) follows from item (v), (33) and the mean value theorem. ∎

5.3. Estimates for ${\mathcal{R}}_{n,\nu}$

Recall the definition of ${\mathcal{Q}}_{n,\nu}$ and ${\mathcal{R}}_{n,\nu}$ in (28). It follows from Lemma 14 that there exists $C_{0}=C_{0}(p,\psi)>0$ such that

[TABLE]

for every $\omega\in\mathbb{R}^{4}$ with $|\omega|\leqslant 5n$, and all $n\in{\mathbb{N}}_{+}$ and $\nu\in(0,1]$.

Theorem 16**.**

Let $\nu\in(0,1]$. There exists $C_{1}=C_{1}(p,A,B,\psi)>0$, not depending on $\nu$, such that ${\mathcal{R}}_{n,\nu}$ is $(A,B)$-convex in $\mathbb{R}^{4}$ for all $n\in{\mathbb{N}}_{+}$. Moreover, the following statements hold.

1. (i)

$D^{2}{\mathcal{R}}_{n,\nu}\in L^{\infty}(\mathbb{R}^{4};\mathbb{R}^{4\times 4})$. 2. (ii)

We have

[TABLE]

pointwise in $\mathbb{R}^{4}$ as $n\rightarrow\infty$. 3. (iii)

For any $n\in{\mathbb{N}}_{+}$ there exists $C(n,\nu,C_{1})>0$ such that

[TABLE] 4. (iv)

There exists $C=C(\nu)>0$ that does not depend on $n$ such that

[TABLE]

for all $\omega\in\mathbb{R}^{4}$, $n\in{\mathbb{N}}_{+}$ and $\nu\in(0,1]$. 5. (v)

For any $n\in{\mathbb{N}}_{+}$ and $\nu>0$ we have

[TABLE]

for all $\zeta,\eta\in\mathbb{R}^{2}$.

Proof.

The $(A,B)$-convexity in the region $\{|\omega|<3n\}\cup\{|\omega|>4n\}$ follows, for any $C_{1}>0$, from the $(A,B)$-convexity of ${\mathcal{Q}}\star\varphi_{\nu}$ and ${\mathcal{P}}_{n}*\varphi_{\nu}$; see Proposition 13 and the first part of Proposition 15 (ii). In order to achieve $(A,B)$-convexity in the region $\{3n\leqslant|\omega|\leqslant 4n\}$, we choose $C_{1}$ large enough and combine (34) with the second part of Proposition 15 (ii).

Item (i) follows from Proposition 15 (v) and the fact that ${\mathcal{Q}}_{n,\nu}\in C^{2}_{c}(\mathbb{R}^{4})$ (or from (34)).

Item (ii) is a trivial consequence of Proposition 15 (i) and the definition of ${\mathcal{Q}}_{n,\nu}$.

From (31) and the fact that $\psi_{n}\equiv 1$ in a neighbourhood of [math], we conclude that $(D{\mathcal{Q}}_{n,\nu})(0)=0$. Hence, by the mean value theorem and the fact that ${\mathcal{Q}}_{n,\nu}\in C_{c}^{\infty}(\mathbb{R}^{4})$, we get $\left|{(D{\mathcal{Q}}_{n,\nu})(\omega)}\right|\leqslant C(\nu,n)|\omega|$. Item (iii) follows from here and Proposition 15 (iv).

Item (iv) follows by combining Lemma 14 (i) and (ii) with Proposition 15 (iii). In particular, use the fact that $D\psi_{n}\equiv 0$ on $\{\omega:|\omega|\not\in[3n,4n]\}$, while, by Lemma 14 (i), on $\{\omega:|\omega|\in[3n,4n]\}$ we have the estimate

[TABLE]

Finally, $1\leqslant|\omega|^{p-1}+|\omega|^{q-1}$, because $|\omega|>1$.

To prove item (v) just observe that ${\mathcal{R}}_{n,\nu}$ is smooth and even in each of the variables $\zeta_{1},\zeta_{2},\eta_{1}$ and $\eta_{2}$, because both ${\mathcal{Q}}\star\varphi_{\nu}$ and ${\mathcal{P}}_{n}\star\varphi_{\nu}$ have this property. ∎

6. Proof of the bilinear embedding (Theorem 2)

As we annunced in Sections 2 and 3, to prove Theorem 2 we modify the heat-flow-Bellman method of [CD-DivForm] by means of the sequence $\{{\mathcal{R}}_{n,\nu}\}$ of Theorem 16.

Let $\Omega\subseteq\mathbb{R}^{d}$ be open. Fix two closed subspaces ${\mathscr{V}}$ and ${\mathscr{V}}^{\prime}$ of $W^{1,2}(\Omega)$ of the type discussed in Section 1.1. Instead of proving (8) directly, it is more convenient to show that

[TABLE]

for all $f,g\in(L^{p}\cap L^{q})(\Omega)$. Once (35) is proved, (8) follows by replacing $f$ and $g$ in (35) with $sf$ and $s^{-1}g$ and minimising the right-hand side with respect to $s>0$.

We first discuss analyticity of the semigroups in (8). Recall the notation $q=p/(p-1)$.

Lemma 17**.**

Let $p\geqslant 2$ and $A\in{\mathcal{A}}(\Omega)$. Suppose that $\Delta_{p}(A)>0$. Then there exists $\vartheta_{0}=\vartheta_{0}(p)\in(0,\pi/2)$ such that $(T^{A,{\mathscr{V}}}_{t})_{t>0}$ is analytic and contractive in $L^{r}(\Omega)$ in the cone

[TABLE]

for all $r\in[q,p]$.

Proof.

By complex interpolation it would be sufficient to prove the statement for $r=p,q$, but we prefer to avoid interpolation and prove the lemma directly for all $r$.

By Lemma 4 (iv), (i) and (iii) there exists $\vartheta_{0}=\vartheta_{0}(p)>0$ such that $\Delta_{r}(e^{i\vartheta}A)>0$ for all $\vartheta\in[-\vartheta_{0},\vartheta_{0}]$ and all $r\in[q,p]$. The contractivity now follows from Proposition 1 and the relation $T^{e^{i\vartheta}A,{\mathscr{V}}}_{t}=T^{A,{\mathscr{V}}}_{e^{i\vartheta}t}$. Finally, analyticity is a consequence of a standard argument [EN, Chapter II, Theorem 4.6]. ∎

Remark 18**.**

In the statement of Lemma 17 we can take any $\vartheta_{0}$ with $\Delta_{p}(e^{\pm i\vartheta_{0}}A)>0$.

For proving (35) we also need the following result that should be compared with [Egert2018, Lemma 4]. Note that here the chain-rule is not a problem, because ${\mathcal{R}}_{n,\nu}$ is smooth.

Lemma 19**.**

Let $u\in{\mathscr{V}}$ and $v\in{\mathscr{V}}^{\prime}$. Then

[TABLE]

for all $n\in{\mathbb{N}}_{+}$ and $\nu>0$.

Proof.

We prove the lemma in the case when ${\mathscr{V}}=W^{1,2}_{D}(\Omega)$ and ${\mathscr{V}}^{\prime}=W^{1,2}_{D^{\prime}}(\Omega)$, for two closed subsets $D,D^{\prime}\subseteq\partial\Omega$. The other cases are simpler and will not be written down here.

Define $\varphi:=(\partial_{\zeta}{\mathcal{R}}_{n,\nu})(u,v)$ and $\psi:=(\partial_{\eta}{\mathcal{R}}_{n,\nu})(u,v)$. Let $u_{\ell}\in C^{\infty}_{c}(\mathbb{R}^{d}\setminus D)$ and $v_{\ell}\in C^{\infty}_{c}(\mathbb{R}^{d}\setminus D^{\prime})$ be such that ${u_{\ell}}_{|_{\Omega}}\rightarrow u$ and ${v_{\ell}}_{|_{\Omega}}\rightarrow v$ in $W^{1,2}(\Omega)$ as $\ell\rightarrow\infty$. Set $\varphi_{\ell}:=(\partial_{\zeta}{\mathcal{R}}_{n,\nu})(u_{\ell},v_{\ell})$ and $\psi_{\ell}:=(\partial_{\eta}{\mathcal{R}}_{n,\nu})(u_{\ell},v_{\ell})$. By Theorem 16 (v) we have ${\rm supp}(\varphi_{\ell})\subseteq\mathbb{R}^{d}\setminus D$ and ${\rm supp}(\psi_{\ell})\subseteq\mathbb{R}^{d}\setminus D^{\prime}$ so, since ${\mathcal{R}}_{n,\nu}$ is smooth, we have $\varphi_{\ell}\in C^{\infty}_{c}(\Omega\setminus D)$ and $\psi_{\ell}\in C^{\infty}_{c}(\Omega\setminus D^{\prime})$. To conclude the proof we now proceed much as in [Egert2018, Lemma 4], but with the simplification that here we can use the chain-rule for the composition of smooth functions. It follows from Theorem 16 (i) and the mean value theorem that

[TABLE]

Therefore $\varphi_{\ell}\rightarrow\varphi$ and $\psi_{\ell}\rightarrow\psi$ in $L^{2}(\Omega)$. Also, by the chain-rule and Theorem 16 (i), the sequence $(\varphi_{\ell})_{\ell\in{\mathbb{N}}_{+}}$ is bounded in $W^{1,2}_{D}(\Omega)$ and the sequence $(\psi_{\ell})_{\ell\in{\mathbb{N}}_{+}}$ is bounded in $W^{1,2}_{D^{\prime}}(\Omega)$. Hence they admit two subsequences weakly convergent in $W^{1,2}_{D}(\Omega)$ and $W^{1,2}_{D^{\prime}}(\Omega)$, respectively. It follows that $\varphi\in W^{1,2}_{D}(\Omega)$ and $\psi\in W^{1,2}_{D^{\prime}}(\Omega)$. ∎

6.1. Proof of (35)

By Lemma 4 (i), we have $\Delta_{p}(A,B)=\Delta_{q}(A,B)$, so it suffices to prove (35) when $p\geqslant 2$. Fix $p\geqslant 2$ and $A,B\in{\mathcal{A}}(\Omega)$ such that $\Delta_{p}(A,B)>0$.

Let ${\mathcal{Q}}={\mathcal{Q}}_{\delta}$ as in (10). Fix $\delta>0$ such that Theorem 6 holds true. Let $\{{\mathcal{P}}_{n}\}_{\in{\mathbb{N}}_{+}}$ be the sequence of Section 4.4. For $n\in{\mathbb{N}}_{+}$ and $\nu\in(0,1]$, define ${\mathcal{R}}_{n,\nu}$ by means of (28) and fix $C_{1}>0$ not depending on $\nu$ such that Theorem 16 holds true.

We now start the heat-flow method of Section 2.4, but for simplicity we omit the subscript ${\mathcal{W}}$. Fix $f,g\in(L^{p}\cap L^{q})(\Omega)$. Define

[TABLE]

The estimates (11) and the analyticity of $(T^{A,{\mathscr{V}}}_{t})_{t>0}$ and $(T^{B,{\mathscr{V}}^{\prime}}_{t})_{t>0}$ (see Lemma 17) imply that ${\mathcal{E}}$ is well defined, continuous on $[0,\infty)$, differentiable on $(0,\infty)$ with a continuous derivative and

[TABLE]

Integrating in the variable $t$ from [math] to $+\infty$ both sides of the equality above, using the first estimate in (11) and the fact that, by analyticity, $T^{A,{\mathscr{V}}}_{t}f\in{\rm D}({\mathscr{L}}^{A}_{p})\cap{\rm D}({\mathscr{L}}^{A}_{q})$ and $T^{B,{\mathscr{V}}^{\prime}}_{t}g\in{\rm D}({\mathscr{L}}^{B}_{p})\cap{\rm D}({\mathscr{L}}^{B}_{q})$, we deduce that for proving (35) it suffices to show that

[TABLE]

for all $u\in{\rm D}({\mathscr{L}}^{A}_{p})\cap{\rm D}({\mathscr{L}}^{A}_{q})$ and all $v\in{\rm D}({\mathscr{L}}^{B}_{p})\cap{\rm D}({\mathscr{L}}^{B}_{q})$.

Note that

[TABLE]

and

[TABLE]

Therefore for proving (36) it suffices to assume that $u\in{\mathscr{V}}$, $v\in{\mathscr{V}}^{\prime}$ and $u,v,{\mathscr{L}}^{A}u,{\mathscr{L}}^{B}v\in L^{p}(\Omega)\cap L^{q}(\Omega)$. By using Theorem 16 (ii) and (iv), Lemma 14 (ii), the fact that ${\mathcal{Q}}\in C^{1}(\mathbb{R}^{4})$ and Lebesgue’s dominated convergence theorem twice, we deduce that

[TABLE]

By Lemma 19 we have $(\partial_{\zeta}{\mathcal{R}}_{n,\nu})(u,v)\in{\mathscr{V}}$ and $(\partial_{\eta}{\mathcal{R}}_{n,\nu})(u,v)\in{\mathscr{V}}^{\prime}$. Hence we can integrate by parts the integral on the right-hand side of (37) and, by means of the chain-rule for the composition of smooth functions with vector-valued Sobolev functions, deduce that

[TABLE]

By Theorem 16, the function ${\mathcal{R}}_{n,\nu}$ is $(A,B)$-convex in $\mathbb{R}^{4}$, so the integrand on the right-hand side of (38) is nonnegative for all $n\in{\mathbb{N}}_{+}$. Hence, by Fatou’s lemma, Theorem 16 (ii) and Proposition 13,

[TABLE]

for all $\nu\in(0,1)$, where $C_{0}=C_{0}(p,\Delta_{p}(A,B),\lambda(A,B),\Lambda(A,B))>0$ does not depend on $\nu$. The desired inequality (36) now follows from (37).

6.2. Remark on the special case $\Omega=\mathbb{R}^{d}$

In this section we simplify the proof of [CD-DivForm, Theorem 1.1] by means of elliptic regularity theory [AgDoNi] and a reduction argument in the spirit of [CD-OU, Section 7].

Let $p>1$ and $q=p/(p-1)$. By the regularisation trick explained in [CD-DivForm, Lemma A4 and Lemma A5], we may assume that $A,B\in C^{\infty}(\mathbb{R}^{d};{\mathbb{C}}^{d\times d})$ with bounded derivatives. In this case, by elliptic regularity [AgDoNi] the semigroups $(T^{A}_{t})_{t>0}$ and $(T^{B}_{t})_{t>0}$ are analytic in $L^{r}(\mathbb{R}^{d})$ and ${\rm D}({\mathscr{L}}^{A}_{r})={\rm D}({\mathscr{L}}^{B}_{r})=W^{2,r}(\mathbb{R}^{d})$, for all $r\in(1,\infty)$; see [Lunardib1, Theorem 3.1.1 and Theorem 3.2.2], [KuW, Section 6], and [pazy, Chapter 7].

Fix $f,g\in(L^{p}\cap L^{q})(\mathbb{R}^{d})$ and start the heat-flow method as in Section 6. Since $T^{A}_{t}f,T^{B}_{t}g\in(W^{2,p}\cap W^{2,q})(\mathbb{R}^{d})$ for all $t>0$, for proving (7) it suffices to show that

[TABLE]

for all $u,v\in(W^{2,p}\cap W^{2,q})(\mathbb{R}^{d})$.

Take $u_{n},v_{n}\in C^{\infty}_{c}(\mathbb{R}^{d})$ such that $u_{n}\rightarrow u$ in $W^{2,p}(\mathbb{R}^{d})={\rm D}({\mathscr{L}}^{A}_{p})$ and $v_{n}\rightarrow v$ in $W^{2,q}(\mathbb{R}^{d})={\rm D}({\mathscr{L}}^{B}_{q})$. By (11) the sequence $(\partial_{\zeta}{\mathcal{Q}})(u_{n},v_{n})$ is bounded in $L^{q}(\mathbb{R}^{d})$ and the sequence $(\partial_{\eta}{\mathcal{Q}})(u_{n},v_{n})$ is bounded in $L^{p}(\mathbb{R}^{n})$. By passing to subsequences, we may assume that $u_{n}\rightarrow u$ and $v_{n}\rightarrow v$ almost everywhere in $\mathbb{R}^{d}$, so that $(\partial_{\zeta}{\mathcal{Q}})(u_{n},v_{n})\rightarrow(\partial_{\zeta}{\mathcal{Q}})(u,v)$ weakly in $L^{q}(\mathbb{R}^{d})$ and $(\partial_{\eta}{\mathcal{Q}})(u_{n},v_{n})\rightarrow(\partial_{\eta}{\mathcal{Q}})(u,v)$ weakly in $L^{p}(\mathbb{R}^{d})$. It follows that it suffices to prove (39) for all $u,v\in C^{\infty}_{c}(\mathbb{R}^{d})$ (alternatively, one can arrive at the very same conclusion by using the analogue of [CD-OU, Lemma 29] which is obtained by replacing $|G|+1$ with $|G|+\gamma$, $\gamma(x)=e^{-|x|^{2}}$).

Fix $u,v\in C^{\infty}_{c}(\mathbb{R}^{d})$. Recall that we are assuming that $A,B$ are smooth. By Lemma 14 (ii) and Lebesgue convergence theorem,

[TABLE]

A standard integration by parts and Proposition 13 now give

[TABLE]

as required for finishing the proof.

7. Maximal regularity and functional calculus: proof of Theorem 3

The interested reader should consult the monographs [KuW, DHMP] and [Haase] for a detailed discussion on the maximal regularity problem for generators of analytic semigroups on Banach spaces; below we shortly describe the problem and recall the principal results we need for proving Theorem 3.

7.1. Maximal regularity

Let ${\mathfrak{X}}$ be a complex Banach space and $-{\mathscr{A}}$ the generator of a strongly continuous semigroup on ${\mathfrak{X}}$. Let $\tau>0$ and $r\in(1,+\infty)$.

We say that ${\mathscr{A}}$ has maximal $L^{r}$-regularity in $(0,\tau)$ if for every $v\in L^{r}(0,\tau;{\mathfrak{X}})$ the unique mild solution

[TABLE]

to the Cauchy problem $u^{\prime}(t)+{\mathscr{A}}u(t)=v(t)$, $u(0)=0$ belongs to $W^{1,r}(0,\tau;{\mathfrak{X}})\cap L^{r}(0,\tau;{\rm D}({\mathscr{A}}))$. This property does not depend on $\tau>0$ and $r>1$ [Dore, Theorem 2.5]. We say that ${\mathscr{A}}$ has (parabolic) maximal regularity if for some, equivalently all, $r>1$ and $\tau>0$ the operator ${\mathscr{A}}$ has maximal $L^{r}$-regularity in $(0,\tau)$. It follows from the very definition that ${\mathscr{A}}$ has maximal regularity if and only if $\sigma I+{\mathscr{A}}$ has maximal regularity for all $\sigma\geqslant 0$; see [Dore, p. 29]. Also, if ${\mathscr{A}}$ has maximal regularity, then there exists $\sigma\geqslant 0$ such that $\exp(-t(\sigma+{\mathscr{A}}))_{t>0}$ is bounded and analytic in ${\mathfrak{X}}$; see [Dore, Theorem 2.2].

Suppose that $-{\mathscr{A}}$ is the generator of a bounded analytic semigroup on a reflexive Banach space ${\mathfrak{X}}$, that is to say, assume that ${\mathscr{A}}$ is sectorial with sectoriality angle $\omega({\mathscr{A}})<\pi/2$ [CDMY]. Denote respectively by ${\rm N}({\mathscr{A}})$ and ${\rm R}({\mathscr{A}})$ the nullspace and the range of ${\mathscr{A}}$. By [CDMY, Theorem 3.8], we have

[TABLE]

where the sum is direct.

As a consequence, we can always factor off the nullspace of ${\mathscr{A}}$ and study maximal regularity for $v\in L^{r}(0,\tau;{\mathfrak{X}})$ such that $v(s)\in\overline{\rm R}({\mathscr{A}})$, for a.e. $s\in(0,\tau)$.

7.2. Functional calculus

Consider a reflexive complex Banach space ${\mathfrak{X}}$ and the generator $-{\mathscr{A}}$ of a bounded analytic semigroup on ${\mathfrak{X}}$. By (41), the restriction ${\mathscr{A}}_{||}$ of ${\mathscr{A}}$ to $\overline{{\rm R}}({\mathscr{A}})$ is a densely defined one-to-one sectorial operator with dense range on the Banach space $\overline{{\rm R}}({\mathscr{A}})$, with sectoriality angle $\omega({\mathscr{A}})<\pi/2$ and the functional calculus introduced in [CDMY] is applicable to it. In particular, for every $\vartheta\in(\omega({\mathscr{A}}),\pi)$ and every bounded and holomorphic function $m$ in the cone $\mathbf{S}_{\vartheta}=\{z\in{\mathbb{C}}\setminus\{0\}:|\arg(z)|<\vartheta|\}$ (in short, for every $m\in H^{\infty}(\mathbf{S}_{\vartheta})$) we may define the closed densely defined, but possibly unbounded, linear operator $m\left({\mathscr{A}}_{||}\right)$. We refer the interested reader to [Mc, CDMY, Haase] for an exhaustive treatment of this subject.

Let $\vartheta\in(\omega({\mathscr{A}}),\pi)$. We say that ${\mathscr{A}}$ admits a bounded $H^{\infty}(\mathbf{S}_{\vartheta})$-calculus if $m({\mathscr{A}}_{||})$ is bounded on $\overline{\rm R}({\mathscr{A}})$ whenever $m\in H^{\infty}(\mathbf{S}_{\vartheta})$. We say that ${\mathscr{A}}$ has a bounded $H^{\infty}$-calculus if it has a bounded $H^{\infty}(\mathbf{S}_{\vartheta})$-calculus for some $\vartheta>\omega({\mathscr{A}})$. The functional calculus angle $\omega_{H^{\infty}}({\mathscr{A}})$ is, by definition, the infimum over all angles $\vartheta>0$ such that ${\mathscr{A}}$ has a bounded $H^{\infty}(\mathbf{S}_{\vartheta})$-calculus (with the convention that $\omega_{H^{\infty}}({\mathscr{A}})=+\infty$ if ${\mathscr{A}}$ does not have a bounded $H^{\infty}$-calculus).

It is an interesting and widely studied problem whether a sectorial operator on a Banach space has a bounded $H^{\infty}$-calculus, and it is of interest to explicitly determine or estimate the functional calculus angle of the operator; see [cowling, Mc, McY, CDMY, KW, KuW, Kalton], [CD-mult, CD-OU] and the references contained therein.

In the special case when ${\mathfrak{X}}={\mathfrak{H}}$ is a Hilbert space, by a universal result of McIntosh [Mc] we always have $\omega_{H^{\infty}}({\mathscr{A}})=\omega({\mathscr{A}})$. The norm of $m({\mathscr{A}})$, $m\in H^{\infty}(\mathbf{S}_{\vartheta})$, $\vartheta>\omega({\mathscr{A}})$, may depend on $\vartheta$, the space ${\mathfrak{H}}$ and the operator ${\mathscr{A}}$. However, by a universal result of Crouzeix and Delyon [CrDe], it is always bounded above by $(2+2/\sqrt{3}){\left\|{m}\right\|_{\infty}}$ whenever $m\in H^{\infty}(\mathbf{S}_{\vartheta})$ and $\vartheta>\sup\left\{\arg(\left\langle{\mathscr{A}}u,u\right\rangle):u\in{\rm D}({\mathscr{A}})\right\}$ (the numerical range angle of ${\mathscr{A}}$).

One reason for studying the boundedness of $H^{\infty}$-calculus for sectorial operators on Banach spaces is its close tie with the maximal regularity problem.

Recall Lemma 17. In the context of the present paper, by either using the Dore-Venni theorem [DoreVenni] in the refined form of Prüss and Sohr [PrussSohr] (see also [GigaSohr]), or the characterisation of maximal regularity by Weis [Weis1] together with the theory developed by Kalton and Weis in [KW], we obtain the following result.

Proposition 20**.**

Let $W^{1,2}_{0}(\Omega)\subseteq{\mathscr{V}}\subseteq W^{1,2}(\Omega)$ as in Section 1.1. Suppose that $A\in{\mathcal{A}}(\Omega)$, $p>1$ and $\Delta_{p}(A)>0$. Let ${\mathscr{L}}^{A}_{p}$ denote the negative generator of $(T^{A,{\mathscr{V}}}_{t})_{t>0}$ in $L^{p}(\Omega)$. If $\omega_{H^{\infty}}({\mathscr{L}}^{A}_{p})<\pi/2$, then ${\mathscr{L}}^{A}_{p}$ has parabolic maximal regularity.

7.3. Proof of Theorem 3

Let $p>1$ and $A\in{\mathcal{A}}(\Omega)$. In light of Proposition 20 it suffices to show that

[TABLE]

By Lemma 4 (i), ii) and a standard duality argument we may assume that $p\geqslant 2$.

By Lemma 4 (ii), (iv) there exists $\vartheta\in(0,\pi/2)$ such that $\Delta_{p}\left(e^{\pm i\vartheta}A,e^{\mp i\vartheta}A^{*}\right)>0$. Then, by Remark 18, for every $r\in[q,p]$ both $(T^{A,{\mathscr{V}}}_{t})_{t>0}$ and $(T^{A^{*}{\mathscr{V}}}_{t})_{t>0}$ are analytic (and contractive) in $L^{r}(\Omega)$ in the cone $\mathbf{S}_{\vartheta}$.

Moreover, by Theorem 2 there exists $C>0$ such that

[TABLE]

for all $f,g\in(L^{p}\cap L^{q})(\Omega)$.

It follows from (42) and the inequality

[TABLE]

that

[TABLE]

for all $f,g\in(L^{p}\cap L^{q})(\Omega)$. Analyticity of $(T^{A,{\mathscr{V}}}_{t})_{t>0}$ in $L^{p}(\Omega)$, Fatou’s lemma and a density argument show that

[TABLE]

for all $f\in L^{p}(\Omega)$ and all $g\in L^{q}(\Omega)$.

We now apply [CDMY, Theorem 4.6 and example 4.8] to the dual subpair $\left\langle\overline{\rm R}({\mathscr{L}}^{A}_{p}),\overline{\rm R}({\mathscr{L}}^{A^{*}}_{q})\right\rangle$ and the dual operators $({\mathscr{L}}^{A}_{p})_{||}$, $({\mathscr{L}}^{A^{*}}_{q})_{||}$, and deduce from (43) that $\omega_{H^{\infty}}({\mathscr{L}}^{A}_{p})\leqslant\pi/2-\vartheta$.

Appendix A Comparison with known results

Under stronger assumptions on $A$ and/or $\Omega$ than those of Theorem 3 it is known that $(T^{A,{\mathscr{V}}}_{t})_{t>0}$, where ${\mathscr{V}}$ is one of the subspaces of Section 1.1, extrapolates to a bounded strongly continuous semigroup on $L^{p}(\Omega)$ in a range of $p$’s larger than the range given by $p$-ellipticity, and the negative generator ${\mathscr{L}}^{A}_{p}$ has parabolic maximal regularity in this larger range of exponents.

A.1. Semigroup extrapolation

Let ${\mathscr{V}}$ denote one of the subspaces of Section 1.1.

(i) For every $\Omega\subseteq\mathbb{R}^{d}$ and every real-valued $A\in{\mathcal{A}}(\Omega)$ the semigroup $(T^{A,{\mathscr{V}}}_{t})_{t>0}$ is sub-Markovian (see [Ouh1992, Ouh1996] and [O, Corollary 4.3 and Corollary 4.10]), so $\omega_{H^{\infty}}({\mathscr{L}}^{A}_{p})<\pi/2$ for all $p>1$; see [cowling], [KS], [CD-mult] for symmetric real-valued $A$ and [KW, Corollary 5.2] for nonsymmetric real-valued $A$. It follows from Dore-Venni theorem [DoreVenni, PrussSohr] that, in this case, ${\mathscr{L}}^{A}_{p}$ has parabolic maximal regularity for all $p>1$.

(ii) As for complex-valued $A\in{\mathcal{A}}(\Omega)$, define the upper and lower Sobolev exponent by the rule $2^{*}=2d/(d-2)$ if $d\geqslant 3$ and $2^{*}=+\infty$ if $d=1,2$; $2_{*}=(2^{*})^{\prime}$. For $A\in{\mathcal{A}}(\Omega)$, $\delta\geqslant 0$ and $\vartheta\in[0,\pi/2)$ denote by $J(A,{\mathscr{V}},\delta,\vartheta)$ the maximal open interval in $(1,+\infty)$ such that $(e^{-\delta z}T^{A,{\mathscr{V}}}_{z})_{z\in{\bf S}_{\vartheta}}$ is uniformly bounded in $L^{p}(\Omega)$, for all $p\in J(A,{\mathscr{V}},\delta,\vartheta)$. Denote by $\omega({\mathscr{L}}^{A}_{2})$ the sectoriality angle of ${\mathscr{L}}^{A}_{2}$.

• (a)

When $\Omega=\mathbb{R}^{d}$, $d=1,2$ and $A\in{\mathcal{A}}(\mathbb{R}^{d})$ we have $J(A,W^{1,2}(\mathbb{R}^{d}),0,\vartheta)=(1,+\infty)$, for all $\vartheta>\pi/2-\omega({\mathscr{L}}^{A}_{2})$ [AMT]. When $\Omega=\mathbb{R}^{d}$, $d\geqslant 3$ and $A\in{\mathcal{A}}(\mathbb{R}^{d})$ Auscher proved [Auscher1] that there exists $\varepsilon>0$ depending only on dimension and the ellipticity constants of $A$ such that $(2_{*}-\varepsilon,2^{*}+\varepsilon)\subset J(A,W^{1,2}(\mathbb{R}^{d}),0,\vartheta)$, for all $\vartheta>\pi/2-\omega({\mathscr{L}}^{A}_{2})$.

• (b)

The results in (a) are sharp [HMMcl]: for all $d\geqslant 3$ and all $p\in[1,\infty]\setminus[2_{*},2^{*}]$ there exists $A\in{\mathcal{A}}(\mathbb{R}^{d})$ such that $(T^{A}_{t})_{t>0}$ is not bounded on $L^{p}(\mathbb{R}^{d})$.

• (c)

In the case when ${\mathscr{V}}$ has the embedding property (${\rm SE}_{q}$) with $q=2^{*}$ Egert implicitly222In [Egert] the author also considered systems. The results are stated under geometric assumptions on $\Omega$ which are stronger than (${\rm SE}_{q}$) with $q=2^{*}$ . These stronger assumptions are used, for example, for proving results on Riesz transforms. However, for the specific result stated here (${\rm SE}_{q}$) with $q=2^{*}$ suffices; see [Egert2018]. proved in [Egert, Theorem 1.6] that $(2_{*},2^{*})\subset J(A,{\mathscr{V}},\delta,\vartheta)$, for every $\delta>0$ and $\vartheta>\pi/2-\omega({\mathscr{L}}^{A}_{2})$. Also, Egert extended the result in (a) ($d\geqslant 3$) by proving that if $\Omega$ is bounded and connected, the boundary is Lipschitz regular around the Neumann part $\overline{\partial\Omega\setminus D}$ and $D$ satisfies further geometric assumptions (see [Egert, H-DJKR, Elst2019]) one always has that $(2_{*}-\varepsilon,2^{*}+\varepsilon)\subset J(A,{\mathscr{V}},\delta,\vartheta)$, for all $\delta>0$ and $\vartheta>\pi/2-\omega({\mathscr{L}}^{A}_{2})$ and some $\varepsilon>0$ depending only on dimension, the ellipticity constants of $A$ and the geometry of $\Omega$. Note that under the above mentioned geometric assumptions there exists a Sobolev extension operator $E:{\mathscr{V}}\rightarrow W^{1,2}(\mathbb{R}^{d})$ and so (${\rm SE}_{q}$) holds true with $q=2^{*}$.

A result similar to (c), but without any further geometric assumption on $D$, has been previously obtained by Tolksdorf [Tolks], who also proved maximal regularity in the range $(2_{*}-\varepsilon,2^{*}+\varepsilon)$.

Recently, Egert improved the result in (c) by combining our notion of $p$-ellipticity and its properties with the technology developed in [Auscher1, Egert]. A similar result has been proved by ter Elst, Haller-Dintelmann, Rehberg and Tolksdorf [Elst2019, Theorem 3.1] by means of a different technique, but still using $p$-ellipticity.

For $p\geqslant 2$ set

[TABLE]

Proposition A.1** ([Egert2018, Theorem 1]).**

Let $d\geqslant 3$. Let $\Omega\subset\mathbb{R}^{d}$ be open. Let ${\mathscr{V}}$ denote one of the subspaces of Section 1.1. Assume the Sobolev embedding (${\rm SE}_{q}$) with $q=2^{*}$. Let $p_{0}>2$ be such that $\Delta_{p_{0}}(A)>0$. Then

[TABLE]

for all $\delta>0$ and $\vartheta>\pi/2-\omega({\mathscr{L}}^{A}_{2})$.

A.2. Functional calculus and maximal regularity

Next result follows by combining Proposition A.1 with a result of Blunck and Kunstmann [BK] that was simplified by Auscher in [Auscher1, Theorem 1.1] and extended to domains $\Omega\subset\mathbb{R}^{d}$ by Egert in [Egert, Proposition 5.2].

Corollary A.2** ([Egert, Egert2018]).**

Under the assumptions of Proposition A.1 we have

[TABLE]

for every $p\in((p^{\circ}_{0})^{\prime},p^{\circ}_{0})$ and $\delta>0$.

As a consequence [DoreVenni, PrussSohr], ${\mathscr{L}}^{A}_{p}$ has parabolic maximal regularity, for every $p\in[(p^{\circ}_{0})^{\prime},p^{\circ}_{0}]$.

A.3. Absence of Sobolev embeddings

It is well-known that (${\rm SE}_{q}$) for $q>2$ requires geometric assumptions on $\Omega$ and does not hold in general, see [AdFour, Theorem 4.46, Theorem 4.48 and Example 4.55] and [BuDa, Proposition 3 and Example 6]. For simplicity, we only discuss the case when ${\mathscr{V}}=W^{1,2}(\Omega)$.

When $\Omega\subset\mathbb{R}^{d}$ has finite measure, by the Rellich-Kondrachov theorem [PioPekka, Theorem 5 and Corollary 1], the Sobolev embedding (${\rm SE}_{q}$) for some $q>2$ implies the compactness of the resolvent operator $(I+{\mathscr{L}}^{A}_{2})^{-1}$ for all $A\in{\mathcal{A}}(\Omega)$ (see also [BuDa, Theorem 7]). So, in this case, the spectrum of ${\mathscr{L}}^{I}_{2}$ is discrete.

1. (a)

A classical example of Courant and Hilbert [CourantHilbert, p. 531] shows that there exists a “rooms and passages” connected bounded domain $\Omega\subset\mathbb{R}^{2}$ for which $0\in\sigma_{\rm ess}({\mathscr{L}}^{I}_{2})$. Actually, for $d\geqslant 2$, given any closed subset $S$ of $[0,+\infty)$ there exists an open connected subset $\Omega$ of the unit ball in $\mathbb{R}^{d}$ such that $\sigma_{\rm ess}({\mathscr{L}}^{I}_{2})=S$, see [HeSeSi]. 2. (b)

Let $d\geqslant 2$. By using a criterion of Evans and Harris [EvHa] (see also [DaSi1, p. 106]), one can easily construct unbounded “horn-shaped” $\Omega\subset\mathbb{R}^{d}$ of finite measure for which the Neumann Laplacian ${\mathscr{L}}^{I}_{2}$ does not have compact resolvent operators. A simple example of this phenomenon is given by the regions

[TABLE]

A.4. Sharpness of Theorem 3

For open sets like those described above in (b), the conclusions of Proposition A.1 and Corollary A.2 are false, because the analyticity angle of the semigroup and the functional calculus angle of the generator may depend on $p$, even for smooth $A$ and pure Neumann boundary conditions.

Kunstmann [kunstmann2] further developed a result of Davies and Simon [DaSi1] for $p=2$ and proved that for $d=2$ and $p>1$ the $L^{p}$ spectrum of the Neumann Laplacian in the region $\Omega_{\alpha}$ satisfies the inclusions

[TABLE]

where

[TABLE]

Moreover, $\sigma_{\rm ess}({\mathscr{L}}^{I}_{p})={\bf P}_{p,\alpha}$.

For $p\in(1,+\infty)$ and $\alpha>0$ the parabolic region ${\bf P}_{p,\alpha}$ is tangent to the critical sector ${\bf S}_{\phi^{*}_{p}}$, where $\phi^{*}_{p}=\arcsin|2/p-1|$. Recall that $\phi_{p}=\pi/2-\phi^{*}_{p}$ and $\phi^{*}_{p}$ are, respectively, the optimal analyticity angle in $L^{p}$ [lp, KR] and the optimal functional calculus angle in $L^{p}$ [CD-mult], for all generators of symmetric contraction semigroups.

By attaching to, say, a ball in $\mathbb{R}^{2}$ countably many disjoint horns $\widetilde{\Omega}_{\alpha_{n}}$, each one congruent to some $\Omega_{\alpha_{n}}$, $\alpha_{n}>0$, Kunstmann [kunstmann2] was able to construct a domain $\Omega_{\max}$ of finite measure such that the associated Neumann Laplacian has maximal $L^{p}$ spectrum:

[TABLE]

Consider the region $\Omega_{{\rm max}}$ described above. Recall from [CD-DivForm, Lemma 5.22] that for $\phi\in(-\pi/2,\pi/2)$ and $p>1$ we have

[TABLE]

Fix $\phi\in(0,\pi/2)$. Since $(T^{I}_{t})_{t>0}$ is symmetric and sub-Markovian, we obtain from [CD-mult], (A.4) and (A.3) that

[TABLE]

Let $p>1$ be such that $\Delta_{p}(e^{i\phi}I)\leqslant 0$, that is, such that $\phi\geqslant\phi_{p}$. Then ${\mathscr{L}}^{e^{i\phi}I}_{p}$ does not have parabolic maximal regularity in $L^{p}(\Omega_{{\rm max}})$, since otherwise, by [Dore, Theorem 2.2], there would exist $\varepsilon,\delta>0$ such that $\sigma({\mathscr{L}}^{I}_{p})\subseteq\overline{\mathbf{S}}_{\pi/2-\phi-\varepsilon}-\delta$, contradicting (A.3).

More generally, by combining [CD-mult] with Proposition 20 and [Kunstmann3, Example 2.4 and Remark 3.2], we deduce that for all $d\geqslant 2$ there exist $\Omega\subset\mathbb{R}^{d}$ and $A\in{\mathcal{A}}(\Omega)$ such that the Neumann operator ${\mathscr{L}}^{A}_{p}$ has parabolic maximal regularity if and only if $\Delta_{p}(A)>0$.

This shows that Theorem 3 is sharp.

A.5. Extrapolation in $L^{\infty}$ for smooth coefficients: counterexamples

Consider the open set $\Omega_{\rm max}$ of [kunstmann2] described above. The equality (A.3) implies that if $\phi>0$ then $(T^{e^{i\phi}I}_{t})_{t>0}$=$(T^{I}_{e^{i\phi}t})_{t>0}$ is not exponentially bounded in $L^{\infty}(\Omega)$. Indeed, assuming the contrary, by interpolation with $L^{2}$ and the relation $\overline{T^{I}_{z}f}=T^{I}_{\bar{z}}\bar{f}$, ${\rm Re}\,z>0$, (because $(T^{I}_{t})_{t>0}$ is positive and analytic in $L^{2}$ in $\{{\rm Re}\,z>0\}$), there would exist $r_{0}>0$ and $\phi_{0}\in(0,\pi/2)$ such that

[TABLE]

This implies that $\sigma({\mathscr{L}}^{I}_{p})\subseteq\overline{{\bf S}}_{\pi/2-\phi_{0}}-r_{0}$ for all $p>2$, contradicting (A.3) when $p$ is such that $\phi^{*}_{p}>\pi/2-\phi_{0}$.

Example A.3**.**

We would expect that if $\phi\neq 0$ then one has $T^{I}_{e^{i\phi}t}(L^{\infty}(\Omega_{\rm max}))\not\subset L^{\infty}(\Omega_{\rm max})$ for some $t>0$, but we could not extract this result from the existing literature. Therefore, we now construct a (disconnected) open set $\Omega\subset\mathbb{R}^{2}$ such that there exists $\phi>0$ for which $T^{e^{i\phi}I}_{1}(L^{\infty}(\Omega))\not\subset L^{\infty}(\Omega)$.

We first consider the Neumann Laplacian $\Delta^{\Omega_{\alpha}}:={\mathscr{L}}^{I}$ in the region $\Omega_{\alpha}$ given by (A.1) with $d=2$, $\alpha>0$ and $c=\alpha^{-1}$. Note that $\Omega_{\alpha}=\alpha^{-1}\cdot\Omega_{1}$. Set $T^{\Omega_{\alpha}}_{t}=\exp(-t\Delta^{\Omega_{\alpha}})$.

By arguing much as in the case of $\Omega_{\rm max}$ discussed above and using (A.2), we see that

[TABLE]

for every $\phi\in(0,\pi/2)$. Fix $\phi\in(0,\pi/2)$. By the uniform boundeness principle, there exists a nonzero $f_{1}\in L^{\infty}(\Omega_{1})$ and a sequence $t_{n}>0$ such that

[TABLE]

We now use a rescaling argument. For $\alpha>0$, consider the operator

[TABLE]

It is not hard to see that

• (i)

$J_{\alpha}:L^{2}(\Omega_{1})\rightarrow L^{2}(\Omega_{\alpha})$ is a surjective isometry with $J^{-1}_{\alpha}=J_{1/\alpha}$;

• (ii)

$J_{\alpha}{\rm D}(\Delta^{\Omega_{1}}_{2})={\rm D}(\Delta^{\Omega_{\alpha}}_{2})$;

• (iii)

$J^{-1}_{\alpha}\Delta^{\Omega_{\alpha}}_{2}J_{\alpha}=\alpha^{2}\Delta^{\Omega_{1}}_{2}$.

It follows that

[TABLE]

Hence, for $\alpha=t_{n}$ and $z=e^{i\phi}$,

[TABLE]

Set $f_{n}=f_{1}(t_{n}\cdot)$. Then ${\left\|{f_{n}}\right\|_{L^{\infty}(\Omega_{t_{n}})}}={\left\|{f_{1}}\right\|_{L^{\infty}(\Omega_{1})}}$ and

[TABLE]

For each $n\in{\mathbb{N}}_{+}$ select a rigid motion of plane $R_{n}$ such that the congruent copies

[TABLE]

are pairwise disjoint. Define

[TABLE]

Then, $\|f\|_{L^{\infty}(\Omega)}={\left\|{f_{1}}\right\|_{L^{\infty}(\Omega_{1})}}$ and

[TABLE]

It follows from (A.5) that $T^{\Omega}_{e^{i\phi}}f\notin L^{\infty}(\Omega)$.

Appendix B Rigidity of generalized convexity

Proposition B.1**.**

Let $A\in{\mathbb{C}}^{d\times d}$ be an elliptic matrix, $t_{0}>0$ and let $\gamma\in C^{1}([0,+\infty);\mathbb{R})\cap C^{2}((0,+\infty)\setminus\{t_{0}\};\mathbb{R})$. Set $\Gamma(\zeta)=\gamma(|\zeta|)$, $\zeta\in\mathbb{R}^{2}$. Suppose that $\Gamma$ is $A$-convex in $\mathbb{R}^{2}\setminus\{|\zeta|=t_{0}\}$. Then,

• (i)

The profile function $\gamma$ is nondecreasing and convex, and $\Gamma$ is convex.

• (ii)

If ${\rm Im}\,A\neq 0$, then either $\gamma={\rm const}$ or $\gamma^{\prime}(t)>0$ for all $t>0$.

Proof.

A rapid calculation shows that

[TABLE]

for all $\zeta\in\mathbb{R}^{2}\setminus(\{|\zeta|=t_{0}\}\cup\{0\})$.

We first prove that $\gamma^{\prime}(t)\geqslant 0$ for all $t>0$. By continuity it suffices to consider $t\neq t_{0}$. For $Z=(Z^{1},Z^{2})\in\mathbb{R}^{d}\times\mathbb{R}^{d}$ write

[TABLE]

Fix $\zeta\in\mathbb{R}^{2}\setminus(\{|\zeta|=t_{0}\}\cup\{0\})$. Take $X^{1}=(x^{1}_{1},0,\dots,0)$, $X^{2}=(x^{2}_{1},0,\dots,0)$. Define $X=(X^{1},X^{2})$,$Y=(Y^{1},Y^{2})={\mathcal{M}}(A)X$, $x=(x^{1}_{1},x^{2}_{1})$ and $y=(y^{1}_{1},y^{2}_{1})$. Then, $\left\langle{\mathcal{M}}(A)X,X\right\rangle_{\mathbb{R}^{2d}}=\left\langle y,x\right\rangle_{\mathbb{R}^{2}}$ and

[TABLE]

Now take $X\neq 0$ of the form above and such that $x$ is orthogonal to $\zeta$. Then, by the ellipticity of $A$, $\left\langle X,{\mathcal{M}}(A)X\right\rangle_{\mathbb{R}^{2d}}\geqslant\lambda|X|^{2}>0$ and by assumption of $A$-convexity of $\Gamma$,

[TABLE]

It follows that $\gamma^{\prime}(|\zeta|)\geqslant 0$.

We now prove that $\gamma$ is convex. It is well-known (and easy to see by means of a convolution argument for regularising $\gamma$) that this is equivalent to proving that $\gamma^{\prime\prime}(t)\geqslant 0$ for all $t\in(0,+\infty)\setminus\{t_{0}\}$. Fix $t\notin\{t_{0},0\}$ and $\zeta\in\mathbb{R}^{2}$ such that $|\zeta|=t$. We rewrite (B.1) as

[TABLE]

where $F_{1}(\zeta)=|\zeta|$.

Therefore, for all $X\in\mathbb{R}^{2d}$ we have

[TABLE]

It follows from (4) that $\Delta_{1}(A)\leqslant 0$. Hence, by Lemma 4 (v) we have

[TABLE]

From this, (B.2), the fact that $\gamma^{\prime}\geqslant 0$, and the inequality $\left\langle X,{\mathcal{M}}(A)X\right\rangle_{\mathbb{R}^{2d}}\geqslant\lambda|X|^{2}$ we deduce that $\gamma^{\prime\prime}(t)\geqslant 0$.

Convexity of $\Gamma$ is now clear and easily follows from the already proved properties of $\gamma$; for the reader’s convenience, we give a complete proof. By using a standard convolution argument, it suffices to prove that $(D^{2}\Gamma)(\zeta)\geqslant 0$, for all $\zeta\notin\{|\zeta|=t_{0}\}\cup\{0\}$. For $\zeta\neq 0$ the two matrices $\tfrac{\zeta}{|\zeta|}\otimes\tfrac{\zeta}{|\zeta|}$ and $I_{\mathbb{R}^{2}}-\tfrac{\zeta}{|\zeta|}\otimes\tfrac{\zeta}{|\zeta|}$ (which represent the orthogonal projections on ${\rm{span}}\{\zeta\}$ and $\zeta^{\bot}$, respectively) are positive semi-definite. Hence $D^{2}\Gamma(\zeta)\geqslant 0$, by (B.1) and the property of $\gamma$ that we have proved above.

We now prove (ii). Suppose that there exists $s>0$ such that $\gamma^{\prime}(s)=0$, then by item (i) $\gamma^{\prime}(t)=0$ for all $t\in[0,s]$. Assume that $\gamma$ is nonconstant. Then,

[TABLE]

$\gamma^{\prime}(t)>0$, for all $t>t_{1}$ and $\gamma^{\prime}(t_{1})=0$. For simplifying the proof, we assume that $t_{1}=t_{0}$. For $t>t_{0}$, define the function $v(t)=\log(\gamma^{\prime}(t))$. Then $\lim_{t\rightarrow t^{+}_{0}}v(t)=-\infty$ and $v^{\prime}(t)=\frac{\gamma^{\prime\prime}(t)}{\gamma^{\prime}(t)}\geqslant 0$, for all $t>t_{0}$. It follows that

[TABLE]

We now show that (B.3) implies ${\rm Im}\,A=0$. Define the function

[TABLE]

Then, by (B.1),

[TABLE]

The formula above expresses the Hessian of the radial function $\Gamma$ in terms of Hessians of power functions. More specifically, by (17) we have

[TABLE]

where

[TABLE]

Since $\Gamma$ is $A$-convex, it follows from the identity above that

[TABLE]

or, equivalently (see Lemma 4 (v)), $\Delta_{p(|\zeta|)}(A)\geqslant 0$, for all $|\zeta|>t_{0}$. Now by (B.3) we have $\sup_{|\zeta|>t_{0}}p(|\zeta|)=+\infty$, so by Lemma 4 (i) and (iii) we have $\Delta_{p}(A)\geqslant 0$, for all $p>1$. Hence, Lemma 4 (vi) implies that ${\rm Im}\,A=0$. ∎

Appendix C Flow regularity

Let $({\mathcal{X}},\mu)$ be a $\sigma$-finite measure space. Fix $p\geqslant 2$ and set $q=p/(p-1)$. Suppose that $(\exp(-t{\mathcal{A}}))_{t>0}$ and $(\exp(-t{\mathcal{B}}))_{t>0}$ are analytic and uniformly bounded both in $L^{p}({\mathcal{X}},\mu)$ and $L^{q}({\mathcal{X}},\mu)$. Let ${\mathcal{Q}}={\mathcal{Q}}_{p,\delta}$ be the Nazarov-Treil Bellman function defined in (10). Fix $f,g\in(L^{p}\cap L^{q})({\mathcal{X}},\mu)$. Consider the flow

[TABLE]

where we omit the subscript ${\mathcal{W}}$ (see Section 2.4).

Proposition C.1**.**

Under the above assumptions, we have:

• (a)

${\mathcal{E}}\in C[0,+\infty)$;

• (b)

${\mathcal{E}}\in C^{1}(0,+\infty)$* and*

[TABLE]

Proof.

We start with (a). We prove only the continuity at [math] since the continuity at other points can be proved exactly in the same way, or it follows from item (b). Set

[TABLE]

By the mean value theorem applied to ${\mathcal{Q}}$,

[TABLE]

Estimates (11) immediately give

[TABLE]

where $C$ does not depend on $x$ and $t$. Now item (a) follows from Hölder’s inequality and the strong continuity of the two semigroups in $L^{p}({\mathcal{X}},\mu)$ and $L^{q}({\mathcal{X}},\mu)$.

We now prove item (b). Analyticity implies that there exist $C>0$ such that, for $r\in\{p,q\}$,

[TABLE]

see, for example, [EN, Chapter II, p.104]. Fix $t_{0}>0$. Then there exists $\delta>0$ such that

[TABLE]

where the series converges in $(L^{p}\cap L^{q})({\mathcal{X}},\mu)$. Moreover,

[TABLE]

and similarly,

[TABLE]

Possibly taking a smaller $\delta$, we also get

[TABLE]

By using the powers series expansion of $e^{-t{\mathcal{A}}}f$ and $e^{-t{\mathcal{B}}}g$ one can also prove that each $e^{-t{\mathcal{A}}}f$ and each $e^{-t{\mathcal{B}}}g$ can be redefined in a set of measure zero, in such a manner that for almost every $x\in\Omega$ the functions $t\mapsto e^{-t{\mathcal{A}}}f(x)$ is real-analytic on $(0,\infty)$, for a.e. $x\in\Omega$; see [stein, p. 72]. Now item (b) follows from estimates (11) and standard theorems of derivation and passage of the limit under the integral sign. ∎

Acknowledgements

The first author was partially supported by the “National Group for Mathematical Analysis, Probability and their Applications” (GNAMPA-INdAM).

The second author was partially supported by the Ministry of Higher Education, Science and Technology of Slovenia (research program Analysis and Geometry, contract no. P1-0291).