Variable Weak Hardy Spaces and Their Applications

TL;DR

This paper introduces variable weak Hardy spaces with multiple characterizations and demonstrates their boundedness under certain Calderón-Zygmund operators, extending classical harmonic analysis to variable exponent settings.

Contribution

It defines variable weak Hardy spaces using maximal functions and establishes their equivalences and boundedness properties under Calderón-Zygmund operators.

Findings

Defined $W ext{H}^{p( ext{·})}( ext{ℝ}^n)$ via radial grand maximal function.

Proved equivalences with atoms, molecules, and Littlewood-Paley functions.

Established boundedness of Calderón-Zygmund operators on these spaces.

Abstract

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-H\"older continuous condition. In this article, the authors first introduce the variable weak Hardy space on $\mathbb R^n$, $W\!H^{p(\cdot)}(\mathbb R^n)$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $W\!H^{p(\cdot)}(\mathbb R^n)$, respectively, by means of atoms, molecules, the Lusin area function, the Littlewood-Paley $g$-function or $g_{\lambda}^\ast$-function. As an application, the authors establish the boundedness of convolutional $\delta$-type and non-convolutional $\gamma$-order Calder\'on-Zygmund operators from $H^{p(\cdot)}(\mathbb R^n)$ to $W\!H^{p(\cdot)}(\mathbb R^n)$ including the critical case $p_-={n}/{(n+\delta)}$, where $p_-:=\mathop\mathrm{ess\,inf}_{x\in \rn}p(x).$