$L^p(\mathbb {R}^d)$ 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

$$\begin{aligned} T_{\Omega ,\,a;k}f(x)=\mathrm{p.\,v.}\int _{\mathbb {R}^d}\frac{\Omega (x-y)}{|x-y|^{d+k}}\big (a(x)-a(y)\big )^kf(y){d}y. \end{aligned}$$

In this paper, the authors prove that if

$$\begin{aligned} \sup _{\zeta \in S^{d-1}}\int _{S^{d-1}}|\Omega (\theta )|\log ^{\beta } \Bigg (\frac{1}{|\theta \cdot \zeta |}\Bigg )d\theta <\infty , \end{aligned}$$

with $\beta \in (1,\,\infty )$, then for $\frac{2\beta }{2\beta -1}<p<2\beta$, $T_{\Omega ,\,a;\,k}$ is bounded on $L^p(\mathbb {R}^d)$.