Multilinear rough singular integral operators

TL;DR

This paper establishes boundedness results for multilinear rough singular integral operators with integrable angular functions, expanding the range of exponents for which these operators are bounded in Lebesgue spaces.

Contribution

It proves boundedness of multilinear rough singular integrals on Lebesgue spaces for a broad set of exponents, including cases where the angular function is only in L^q with q≥2.

Findings

Boundedness of operators on a convex polyhedral set of exponents

Extension to rough kernels with minimal smoothness assumptions

Largest known open set of exponents for these operators

Abstract

We study $m$-linear homogeneous rough singular integral operators $\mathcal{L}_{\Omega}$ associated with integrable functions $\Omega$ on $\mathbb{S}^{mn-1}$ with mean value zero. We prove boundedness for $\mathcal{L}_{\Omega}$ from $L^{p_1}\times \cdots \times L^{p_m}$ to $L^p$ when $1<p_1,\dots, p_m<\infty$ and $1/p=1/p_1+\cdots +1/p_m$ in the largest possible open set of exponents when $\Omega \in L^q(\mathbb S^{mn-1})$ and $q\ge 2$. This set can be described by a convex polyhedron in $\mathbb R^m$.