Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents

Shengrong Wang,Jingshi Xu

doi:10.1515/math-2021-0024

Shengrong Wang, Jingshi Xu

Open Access

https://doi.org/10.1515/math-2021-0024

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Abstract

Abstract If vector-valued sublinear operators satisfy the size condition and the vector-valued inequality on weighted Lebesgue spaces with variable exponent, then we obtain their boundedness on weighted Herz-Morrey spaces with variable exponents.

Highlights

Since the fundamental paper [1] by Kováčik and Rákosník appeared in 1991, the Lebesgue spaces with variable exponent have been extensively studied by many authors; see [2,3,4]
Motivated by applications to fluid dynamics, image restoration and partial differential equations with non-standard growth conditions, many variable spaces were introduced, such as Besov and Triebel-Lizorkin spaces with variable exponents [5,6,7,8,9,10,11,12], Besov-type and Triebel-Lizorkin-type spaces with variable exponents [13,14,15,16,17,18,19,20,21], Hardy spaces with variable exponent [22], Bessel potential spaces with a variable exponent [23,24] and Morrey spaces with variable exponents [25]
After that the theory of these spaces had a remarkable development in part due to its usefulness in applications. They appear in the characterization of multipliers on Hardy spaces [27], in the summability of Fourier transforms [28] and in regularity theory for elliptic equations in divergence form [29]

Summary

Introduction

Since the fundamental paper [1] by Kováčik and Rákosník appeared in 1991, the Lebesgue spaces with variable exponent have been extensively studied by many authors; see [2,3,4]. Herz-Morrey spaces with variable exponents were introduced in [35]. Izuki [35] obtained the boundedness of vector-valued sublinear operators satisfying a size condition on Herz-Morrey spaces with variable exponent MKqα,,pλ(⋅)( n). Wang and Shu [37] obtained the boundedness of some sublinear operators on weighted variable Herz-Morrey spaces MKqα,,pλ(⋅)( n, w). Motivated by the mentioned work, in this paper, we will prove the boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents MKqα((⋅⋅)),,pλ(⋅)(w). We obtain the boundedness of vector-valued Hardy-Littlewood maximal operator on weighted Herz-Morrey spaces with variable exponents MKqα((⋅⋅)),,pλ(⋅)(w). It is well known that the boundedness of vector-valued Hardy-Littlewood maximal operator on non-weighted and weighted Lebesgue spaces play a key role in the theory of function spaces.

Notations and main result

Proof of Theorem 1