Boundedness and Compactness of Multilinear Singular Integrals on Morrey Spaces

Abstract

In this paper, we consider the boundedness and compactness of the multi-linear singular integral operator on Morrey spaces, which is defined by $T_{A} f(x)=\text { p.v. } \int_{\mathbb{R}^{n}} \frac{\Omega(x-y)}{|x-y|^{n+1}} R(A ; x, y) f(y) d y,$ where $R(A ; x, y)=A(x)-A(y)-\nabla A(y) \cdot(x-y)$ with $D^{\beta} A \in B M O\left(\mathbb{R}^{n}\right)$ for all $|\beta|=1$ We prove that T_A is bounded and compact on Morrey spaces $L^{p, \lambda}\left(\mathbb{R}^{n}\right) \text { for all } 1<p<\infty$ with $\Omega \text { and } A$ satisfying some conditions. Moreover, the boundedness and compactness of the maximal multilinear singular integral operator T_A,∗ on Morrey spaces are also given in this paper.

Keywords

Multilinear operator / compactness / rough kernel / Morrey space