Strongly Singular Convolution Operators on Herz-Type Hardy Spaces with Variable Exponent

Hongbin Wang,Dunyan Yan

doi:10.1155/2019/6821464

Abstract

We investigate the boundedness of the strongly singular convolution operators on Herz-type Hardy spaces with variable exponent.

Highlights

We investigate the boundedness of the strongly singular convolution operators on Herz-type Hardy spaces with variable exponent
The theory of function spaces with variable exponents has been extensively studied by researchers since the work of Kovacǐ k and Rakosnık [1] appeared in 1991
7], the authors proved the boundedness of some integral operators on variable function spaces

Summary

Introduction

The theory of function spaces with variable exponents has been extensively studied by researchers since the work of Kovacǐ k and Rakosnık [1] appeared in 1991. In [2, 3] the authors defined the Herz-type Hardy spaces with variable exponent and gave some characterizations for them. In [2], the authors gave the definition of the Herz-type Hardy space with variable exponent HKqα(,⋅p)(Rn) and the atomic decomposition characterizations. (i) The homogeneous Herz-type Hardy space with variable exponent HKqα(,⋅p)(Rn) is defined by HKqα(,⋅p) (Rn) (17). (ii) The nonhomogeneous Herz-type Hardy space with variable exponent HKqα(,⋅p)(Rn) is defined by HKqα(,⋅p) (Rn) (19). Weighted Lq norm and weak(1,1) estimates were established by Chanillo in [13] The boundedness of these operators on the weighted Herz-type. Motivated by [2, 14], we will study the boundedness of the strongly singular convolution operators Tb on Herz-type Hardy spaces with variable exponent.

Preliminary Lemmas

The Proof of Main Results