Necessary and sufficient conditions for the boundedness of rough multilinear fractional operators on Morrey-type spaces

Zhiheng Wang,Zengyan Si

doi:10.1186/s13660-015-0627-2

Abstract

In this paper, we study the necessary and sufficient conditions on the parameters for the boundedness of the multilinear fractional maximal operator $\mathcal{M}_{\Omega,\alpha}$ and the multilinear fractional integral operator $\mathcal{I}_{\Omega,\alpha}$ with rough kernels on Morrey spaces and modified Morrey spaces, respectively. This extends some recent results of Guliyev, Hasnov and Zeren; the necessary and sufficient conditions for the boundedness of $M_{\alpha}$ and $I_{\alpha}$ on modified spaces are considered.

Highlights

Kenig and Stein [ ] studied the boundedness of multilinear fractional integral operator Iα,m, < α < mn, on Lebesgue spaces
We study the necessary and sufficient conditions on the parameters for boundedness of the multilinear fractional maximal operator M,α and the multilinear fractional integrals I,α with rough kernels on Morrey spaces and modified Morrey spaces, respectively, whose definitions are given below
(ii) If p =, the condition – /q = α/(n – λ) is necessary and sufficient for the boundedness of the operator Iα from L,λ(Rn) to WLq,λ(Rn). Motivated by these two results above, we study the necessary and sufficient conditions on the parameters for the boundedness of the multilinear fractional maximal operator M,α and the multilinear fractional integral operator I,α with rough kernels on Morrey spaces and modified Morrey spaces, respectively

Summary

Introduction

Kenig and Stein [ ] studied the boundedness of multilinear fractional integral operator Iα,m, < α < mn, on Lebesgue spaces. We study the necessary and sufficient conditions on the parameters for boundedness of the multilinear fractional maximal operator M ,α and the multilinear fractional integrals I ,α with rough kernels on Morrey spaces and modified Morrey spaces, respectively, whose definitions are given below. (ii) If p = , the condition – /q = α/(n – λ) is necessary and sufficient for the boundedness of the operator Iα from L ,λ(Rn) to WLq,λ(Rn) Motivated by these two results above, we study the necessary and sufficient conditions on the parameters for the boundedness of the multilinear fractional maximal operator M ,α and the multilinear fractional integral operator I ,α with rough kernels on Morrey spaces and modified Morrey spaces, respectively. Since each pj > s , by the Hölder inequality and Lemma . and Lemma . , we have

Rn m

Lpj pj

Then by