On the Multilinear Fractional Integral Operators with Correlation Kernels

TL;DR

This paper investigates multilinear fractional integral operators with correlation kernels, establishing conditions for their boundedness between various Lebesgue spaces and endpoint estimates into BMO.

Contribution

It provides necessary and sufficient conditions for the boundedness of these operators, including endpoint estimates, advancing understanding of their functional analysis properties.

Findings

Derived boundedness conditions for the operators.

Established endpoint estimates into BMO.

Extended classical results to multilinear correlation kernels.

Abstract

In this paper, we study a class of multilinear fractional integral operators which have correlation kernels $\prod_{1\leq i<j \leq k}|x_i-x_j|^{-\alpha_{ij}}$. The necessary and sufficient conditions are obtained under which these oprators are bounded from $L^{p_1}\times \cdots \times L^{p_k}$ into $L^q$. As a consequence, we also get the endpoint estimates from $L^{p_1}\times \cdots \times L^{p_k}$ to $BMO$ of these operators.