Bilinear endpoint estimates for Calder\'on commutator with rough kernel

TL;DR

This paper establishes new bilinear endpoint estimates for Calderón commutators with rough kernels, improving previous results and introducing novel endpoint bounds for related maximal functions in harmonic analysis.

Contribution

The paper provides the first endpoint estimates for Calderón commutators with rough kernels, extending the understanding of their mapping properties in Lebesgue spaces.

Findings

Proves Calderón commutator maps $L^q \times L^1$ to weak $L^r$ for $q>d$.

Establishes new endpoint estimate for $q=d$ case, even with smooth kernels.

Introduces a new endpoint estimate for the Mary Weiss maximal function.

Abstract

In this paper, we establish some bilinear endpoint estimates of Calder\'on commutator $\mathcal{C}[\nabla A,f](x)$ with a homogeneous kernel when $\Omega\in L\log^+L(\mathbf{S}^{d-1})$. More precisely, we prove that $\mathcal{C}[\nabla A,f]$ maps $L^q(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to $L^{r,\infty}(\mathbb{R}^d)$ if $q>d$ which improves previous result essentially. If $q=d$, we show that Calder\'on commutator maps $L^{d,1}(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to $L^{r,\infty}(\mathbb{R}^d)$ which is new even if the kernel is smooth. The novelty in the paper is that we prove a new endpoint estimate of the Mary Weiss maximal function which may have its own interest in the theory of singular integral.