Molecular Decomposition of Anisotropic Hardy Spaces with Variable Exponents

TL;DR

This paper develops a molecular decomposition for variable anisotropic Hardy spaces, extending classical results and enabling new boundedness results for Calderón-Zygmund operators in variable exponent settings.

Contribution

It introduces a molecular decomposition for variable anisotropic Hardy spaces, a novel result even in the classical isotropic case, and applies it to operator boundedness.

Findings

Established molecular decomposition for $H^{p(ullet)}_A(R^n)$

Proved boundedness of Calderón-Zygmund operators on these spaces

Extended classical isotropic results to variable anisotropic setting

Abstract

Let $A$ be an expansive dilation on $\mathbb{R}^n$, and $p(\cdot):\mathbb{R}^n\rightarrow(0,\,\infty)$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition. Let $H^{p(\cdot)}_A({\mathbb {R}}^n)$ be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the authors establish its molecular decomposition, which is still new even in the classical isotropic setting (in the case $A:=2\mathrm I_{n\times n}$). As applications, the authors obtain the boundedness of anisotropic Calder\'on-Zygmund operators from $H^{p(\cdot)}_{A}(\mathbb{R}^n)$ to $L^{p(\cdot)}(\mathbb{R}^n)$ or from $H^{p(\cdot)}_{A}(\mathbb{R}^n)$ to itself.