The Boundedness of Multilinear Commutators on Grand Variable Herz Spaces

--

(1. Business Management Department, Anhui Vocational College of Press and Publishing, Hefei 230601, China; 2. Department of Mathematics and Computer Science, Tongling University, Tongling 244000, China)

Abstract: We consider multilinear commutators of singular integrals defined by

Keywords: Grand Herz space; Variable exponent; Multilinear commutators

§1. Introduction

Let△be the diagonal of Rn×Rn:△={(x,x):x∈Rn}. Following Coifman and Meyer [5],we say thatK:Rn×Rn△→C is a standard kernel if there exist positive constantsγandCsuch that

wherem∈N*. Obviously, whenm=1 this definition coincides with the commutatorTbas in(1.4). Moreover, they proved thatT→bis a bounded operator onLp(ω) for anyω ∈Ap, 1＜p＜∞,whereApdenotes Muckenhoupt’s weight class (see [22] for the definition). Subsequently, many authors made important progress on these commutators, among them we refer to Zhou, Ma and Xu [27], Si and Xue [19] and references therein.

§2. Preliminaries

In this section, we recall some basic facts on Lebesgue spaces with variable exponent. We refer to the monographs [7,9] for more information.

We denote byP(Rn) the set of all measurable functionsp(·):Rn →[1,∞). Forp(·)∈P(Rn),we use the notation

§3. Boundedness on grand variable Herz spaces

This completes the proof of Theorem 3.1.