Abstract
In this paper, our aim is to prove the boundedness of commutators generated by the Marcinkiewicz integrals operator [b,μΩ] and obtain the result with Lipschitz function and BMO function f on the Herz-Morrey-Hardy spaces with variable exponents .
Highlights
PreliminariesWe give some preliminaries which we used to prove theorems. ( ) Lemma 2.1. [11] Let p (⋅) ∈ n
In 1938, Marcinkiewicz [1] introduced the Marcinkiewicz integral
Stain in [2] introduced the Marcinkiewicz integral operator related to the littlewood-Paley g function on n and proved that μΩ is of type ( p, p) for 1 < p ≤ 2 and of week type (1,1)
Summary
We give some preliminaries which we used to prove theorems. ( ) Lemma 2.1. [11] Let p (⋅) ∈ n. [14] Let k be a positive integer and B be a ball in n. [15] Let q (⋅) ∈ B n , there exist positive constants. ΧB ( ) Lq′(⋅) n ≤ C B , where δ1,δ2 are constants with 0 < δ1,δ2 < 1. [16] If q (⋅) ∈ B n , there exists a constant C > 0 such that for any balls B in n ,. [15] Given E, let q (⋅) ∈ ( E ), f : E × E → n be a measurable function (with respect to product measure) such that for almost every y ∈ E, f (., y) ∈ Lq(⋅) ( E ). If Q ≤ 2n and x ∈ Q if Q ≥ 1 for every cube (or ball) Q ∈ n , where p (∞) =limx→∞ p ( x)
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