The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents

Qingdong Guo,Wenhua Wang

doi:10.1515/math-2020-0031

Abstract

Abstract In this article, the authors establish the characterizations of a class of anisotropic Herz-type Hardy spaces with two variable exponents associated with a non-isotropic dilation on ℝ n {{\mathbb{R}}}^{n} in terms of molecular decompositions. Using the molecular decompositions, the authors obtain the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy space with two variable exponents.

Highlights

The theory of function spaces with variable exponents has rapidly made progress in the past 20 years since some elementary properties were established by Kováčik and Rákosník [1]
2012, Almeida and Drihem [3] introduced the Herz spaces with two variable exponents and obtain the boundedness of some sublinear operators on those spaces
In 2003, Bownik [6] introduced the anisotropic Hardy spaces HAp ( n) associated with very general discrete groups of dilations. This formulation includes the classical isotropic Hardy space theory established by Fefferman and Stein [7] and the parabolic Hardy space theory established by Calderón and Torchinsky [8,9]

Summary

Introduction

The theory of function spaces with variable exponents has rapidly made progress in the past 20 years since some elementary properties were established by Kováčik and Rákosník [1]. Wang and Liu [4] introduced the Herz-type Hardy spaces with variable exponents HKpα(,⋅q) In 2003, Bownik [6] introduced the anisotropic Hardy spaces HAp ( n) associated with very general discrete groups of dilations. In 2008, Ding et al [10] introduced the anisotropic Herz-type Hardy spaces HKpα,q (A; n) and HKpα,q (A; n) and established their atomic and molecular decompositions. In 2018, Zhao and Zhou [11] introduced the variable anisotropic Herz-type Hardy spaces HKpα(,⋅q) (A; n) and HKpα,(q⋅) (A; n) and established their atomic and molecular decompositions. Using these decompositions, they gave some applications. HKpα((⋅⋅)),q (A; n) and HKpα((⋅⋅)),q (A; n) and established their atomic decomposition and some applications

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It follows that

Hardy space