Rough Multilinear Fractional Integrals on Weighted Morrey Spaces

Runqing Cui,Xiao Li

doi:10.1155/2015/731487

Abstract

It is showed that a class of multilinear fractional operators with rough kernels, which are similar to the higher-order commutators for the rough fractional integrals, are bounded on the weighted Morrey spaces.

Highlights

Introduction and ResultsSuppose that Sn−1 denotes the unit sphere of Rn (n ≥ 2) equipped with the usual Lebesgue measure, Ω ∈ Ls(Sn−1) (s > 1) is homogeneous of degree zero, and Aj, j = 1, . . . , k, are functions defined on Rn
Consider the following multilinear fractional integral with rough kernel defined by TΩA,1α,...,Ak (f) (x)
In order to deal with the fractional order case, we need to consider the weighted Morrey spaces with two weight functions; they were introduced by Komori and Shirai in [9]

Summary

Introduction and Results

The purpose of this paper is to discuss the boundedness properties of the rough fractional multilinear integral operators TΩA,1α,...,Ak on appropriate weighted Morrey spaces. In order to deal with the fractional order case, we need to consider the weighted Morrey spaces with two weight functions; they were introduced by Komori and Shirai in [9]. Let 1 ≤ p < ∞, φ a positive measurable function on Rn ×(0, ∞), and ω a nonnegative measurable function on Rn. We denote by Mφp(ω, Rn) other weighted Morrey spaces, the spaces of all functions f ∈ Lploc(ω) with finite norm. In Corollary 6, let k = 1 and m1 = 1; we obtain the main result in [11].

Some Preliminaries

A Local Estimate

Proof of Theorem 3