Bilinear endpoint estimates for Calderón commutator with rough kernel

Abstract

In this paper, we establish some bilinear endpoint estimates of Calderón 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 Calderón 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.

11 pages,1 figure