An endpoint estimate for the maximal Calderón commutator with rough kernel

Abstract

In this paper, the authors consider the endpoint estimates for the maximal Calderón commutator defined by $$T_{\hat{I}\text{c◯},a}^{\ast }f(x)=\underset{\hat{I}\mu >0}{\sup }\left|{\int }_{|x-y|>\hat{I}\mu }\frac{\hat{I}\text{c◯}(x-y)}{|x-y{|}^{d+1}}\right(a(x)-a(y))f(y)dy|,$$ where $\hat{I}\text{c◯}$ is homogeneous of degree zero, integrable on $S^{d-1}$ and has vanishing moment of order one, $a$ be a function on ${\mathbb{R}}^d$ such that $\nabla a\in L^{\infty }({\mathbb{R}}^d)$. The authors prove that if $\hat{I}\text{c◯}\in L\log L(S^{d-1})$, then $T_{\hat{I}\text{c◯},a}^{\ast }$ satisfies an endpoint estimate of $L\log \log L$ type.

25 pages