Weighted Morrey Estimates for Multilinear Fourier Multiplier Operators

Abstract

The multilinear Fourier multipliers and their commutators with Sobolev regularity are studied. The purpose of this paper is to establish that these operators are bounded on certain product Morrey spaces $L^{p,k}({ℝ}^n)$. Based on the boundedness of these operators from $L^{p_1}({\omega }_1)\times \cdots \times L^{p_m}({\omega }_m)$ to $L^p({\prod }_{j=1}^m\mathrm{‍}{\omega }^{p/p_j})$, we obtained that they are also bounded from $L^{p_1,k}({\omega }_1)\times \cdots \times L^{p_m,k}({\omega }_m)$ to $L^{p,k}({\prod }_{j=1}^m\mathrm{‍}{\omega }^{p/p_j})$, with , , $1/p=1/p_1+\cdots +1/p_m$, and ${\omega }_j\in A_{p_j},j=1,\dots ,m$.