Estimates for Schr\"{o}dinger Groups and Imaginary Power Operators on Weak Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces

TL;DR

This paper establishes boundedness estimates for Schrödinger groups and imaginary power operators on weak Hardy spaces linked to non-negative self-adjoint operators within a broad class of function spaces on metric measure spaces.

Contribution

It introduces new boundedness results for Schrödinger groups and imaginary powers on weak Hardy spaces associated with general operators and function spaces, extending prior work to more general settings.

Findings

Boundedness of Schrödinger groups on weak Hardy spaces.

Boundedness of imaginary power operators on weak Hardy spaces.

Applicability to various function spaces including weighted and variable Lebesgue spaces.

Abstract

Let $(\mathbb{X},d,\mu)$ be a doubling metric measure space, $L$ a non-negative self-adjoint operator on $L^2(\mathbb{X})$ satisfying the Davies-Gaffney estimate, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$ satisfying some mild assumptions with $p\in(0,\infty)$ and $s_0\in(0,\min\{p,1\}]$. In this article, the authors study the weak Hardy space $WH_{X,L}(\mathbb{X})$ associated with $L$ and $X(\mathbb{X})$, and then give the atomic and molecular decompositions of $WH_{X,L}(\mathbb{X})$. As applications, the authors establish the boundedness estimate of Schr\"{o}dinger groups for fractional powers of $L$ on $WH_{X,L}(\mathbb{X})$: $$\left\|(I+L)^{-\beta/2}e^{i\tau L^{\gamma/2}}f\right\|_{WH_{X,L}(\mathbb{X})}\leq C\left(1+|\tau|\right)^{n(\frac{1}{s_0}-\frac{r}{2})}\|f\|_{WH_{X,L}(\mathbb{X})},$$ where $0<\gamma\neq1$, $\beta\in[\gamma n(\frac{1}{s_0}-\frac{1}{2}),\infty)$, $r\in(0,1]$, $\tau\in \mathbb{R}$, and $C>0$ is a constant. Moreover, when $(\mathbb{X},d,\mu)$ is an Ahlfors $n$-regular metric measure space and $L$ satisfies the Gaussian upper bound estimate, the authors also obtain the boundedness estimate of imaginary power operators of $L$ on $WH_{X,L}(\mathbb{X})$: $$\left\|L^{i\tau}f\right\|_{WH_{X,L}(\mathbb{X})}\leq C\left(1+|\tau|\right)^{n(\frac{1}{s_0}-\frac{r}{2})}\|f\|_{WH_{X,L}(\mathbb{X})},$$ where $\alpha>n(\frac{1}{s_0}-\frac{1}{2})$, $r\in(\frac{n/s_0}{\alpha+n/2},1]$, $\tau\in \mathbb{R}$, and $C>0$ is a constant. These results are also novelty for strong Hardy spaces $H_{X,L}(\mathbb{X})$. Moreover, all these results have a wide range of generality and, particularly, even when they are applied to weighted Lebesgue spaces, mixed-norm Lebesgue spaces, Orlicz spaces, variable Lebesgue spaces and Euclidean spaces setting, these results are also new.