Boundedness of Marcinkiewicz Integrals and Their Commutators on Generalized Weighted Morrey Spaces

Runqing Cui,Zhang Li

doi:10.1155/2015/450145

Abstract

We will study the boundedness of Marcinkiewicz integrals with rough kernel on the generalized weighted Morrey spaces. We will also prove that the commutator operators formed by aBMORnfunction and Marcinkiewicz integrals are also bounded on the generalized weighted Morrey spaces.

Highlights

Introduction and ResultsSuppose that Sn−1 is the unit sphere in Rn (n ≥ 2) equipped with the normalized Lebesgue measure dσ
Let Ω ∈ Lq(Sn−1) with 1 < q ≤ ∞ be homogeneous of degree zero and satisfy the cancellation condition
The Marcinkiewicz integral of higher dimension μΩ is defined by μΩ

Summary

Introduction and Results

Suppose that Sn−1 is the unit sphere in Rn (n ≥ 2) equipped with the normalized Lebesgue measure dσ. The following results concerning the boundedness of Marcinkiewicz integrals and their commutators on weighted Lp space are known. 󵄩󵄩󵄩󵄩[b, μΩ] (f)󵄩󵄩󵄩󵄩Lp(ω,Rn) ≤ C ‖b‖∗ 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩Lp(ω,Rn). In [9] Mizuhara gave generalization Morrey spaces Lp,Φ(Rn) considering Φ(r) instead of rλ in (7) He studied a continuity in Lp,Φ(Rn) of some classical integral operators. Let 1 ≤ p < ∞ and let φ be a positive measurable function on Rn × (0, ∞) and let ω be a nonnegative measurable function on Rn. Following [12], we denote by Mφp(ω, Rn) the generalized weighted Morrey space, the space of all functions f ∈ Lploc(ω, Rn) with finite norm. The purpose of this paper is to discuss the boundedness properties of Marcinkiewicz integrals with rough kernel and their commutators on the generalized weighted Morrey spaces Mφp(ω, Rn). 󵄩󵄩󵄩󵄩[b, μΩ] f󵄩󵄩󵄩󵄩Lp,Φ(Rn) ≤ C ‖b‖∗ 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩Lp,Φ(Rn)

Some Preliminaries

Proof of Theorem 3

Proof of Theorem 4