Variable Weak Hardy Spaces and Their Applications

Abstract

Let $p(\cdot ):{\mathbb{R}}^n\to (0,\infty )$ be a variable exponent function satisfying the globally log-Hölder 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 Calderón-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 $p_-:=\mathop\mathrm{ess\,inf}_{x\in \rn}p(x).$

This is a modified version of the published version. We only modify Theorems 7.4 and 7.6 a little bit

References in corpus (4)

• Atomic and molecular decompositions in variable exponent 2-microlocal spaces and applications
• Fractional, Maximal and Singular Operators in Variable Exponent Lorentz Spaces
• Intrinsic Square Function Characterizations of Hardy Spaces with Variable Exponents
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Cited by in corpus (6)

• Variable Martingale Hardy Spaces and Their Applications in Fourier Analysis
• Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood--Paley Characterizations and Real Interpolation
• Applications of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces
• Littlewood-Paley Characterizations of Hardy-type Spaces Associated with Ball Quasi-Banach Function Spaces
• Boundedness of Singular Integral Operators on Weak Herz Type Spaces with Variable Exponent
• Real-Variable Characterizations of Orlicz-Slice Hardy Spaces