$L^p({\mathbb{R}}^d)$ boundedness for the Calderón commutator with rough kernel

Abstract

Let $k\in \mathbb{N}$, $\Omega$ be homogeneous of degree zero, integrable on $S^{d-1}$ and have vanishing moment of order $k$, $a$ be a function on ${\mathbb{R}}^d$ such that $\nabla a\in L^{\infty }({\mathbb{R}}^d)$, and $T_{\Omega ,a;k}$ be the $d$-dimensional Calderón commutator defined by $$T_{\Omega ,a;k}f(x)=p.v.{\int }_{{\mathbb{R}}^d}\frac{\Omega (x-y)}{|x-y{|}^{d+k}}(a(x)-a(y){)}^{k}f(y)dy.$$ In this paper, the authors prove that if $$\underset{\zeta \in S^{d-1}}{\sup }{\int }_{S^{d-1}}|\Omega (\theta )|{\log }^{\beta }(\frac{1}{|\theta \cdot \zeta |})d\theta <\infty ,$$ with $\beta \in (1,\infty ]$, then for $\frac{2\beta }{2\beta -1}<p<2\beta$, $T_{\Omega ,a;k}$ is bounded on $L^p({\mathbb{R}}^d)$.

20 pages