Multilinear Operators on Weighted Amalgam-Type Spaces

Abstract: In this paper, we prove that if a multilinear operator $\mathcal{T}$ and its multilinear commutator $\mathcal{T}{\Sigma\vec{b}}$ and iterated commutator $\mathcal{T}{\Pi\vec{b}}$ for $\vec{b}\in(\mathbb{R}^n)^m$ are bounded on product weighted Lebesgue space, then $\mathcal{T}$, $\mathcal{T}{\Sigma\vec{b}}$ and $\mathcal{T}{\Pi\vec{b}}$ are also bounded on product weighted Amalgam space. As its applications, we show that multilinear Littlewood-Paley functions and multilinear Marcinkiewicz integral functions with kernels of convolution type and non-convolution type, and their multilinear commutators and iterated commutators are bounded on product weighted Amalgam space. We also consider multilinear fractional type integral operators and their commutators' behaviors on weighted amalgam space. In order to deal with the endpoint case, we introduce the amalgam-Campanato space and show that fractional integral integral operator are bounded operators from product amalgam space to amalgam-Campanato space. What should point out is that even if in the linear case, our results for fractional integral operator are also new.