Multilinear singular integrals with homogeneous kernels near $L^1$

TL;DR

This paper establishes the optimal bounds for multilinear singular integral operators with homogeneous kernels on product Lebesgue spaces, extending the understanding of their boundedness properties.

Contribution

It determines the precise range of Lebesgue space exponents for which these multilinear operators are bounded, improving previous results.

Findings

Optimal $L^{p_1} imes \

L^{p_m} o L^p$ bounds established

Boundedness depends on the integrability of the kernel function $\

Abstract

We obtain the optimal open range of $L^{p_1}(\mathbb R^n)\times\cdots\times L^{p_m}(\mathbb R^n)\to L^p(\mathbb R^n)$ bounds for multilinear singular integral operators with homogeneous kernels of the form $\Omega(\frac{y}{|y|})|y|^{-mn}$, where $\Omega$ is a function in $L^{q}(\mathbb S^{mn-1})$ with vanishing integral and $q>1$.