Dualities of variable anisotropic Hardy spaces and boundedness of singular integral operators
Bull. Korean Math. Soc. 2021 Vol. 58, No. 2, 365-384
Published online November 5, 2020
Printed March 31, 2021
Wenhua Wang
Wuhan University
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 author obtains the boundedness of anisotropic convolutional $\delta$-type 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. In addition, the author also obtains the duality between $H^{p(\cdot)}_ A(\mathbb{R}^n)$ and the anisotropic Campanato spaces with variable exponents.
Keywords : Anisotropy, Hardy space, atom, Calder\'on-Zygmund operator, Campanato space
MSC numbers : Primary 42B20; Secondary 42B30, 46E30
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