Quantitative weighted bounds for Calderón commutator with rough kernel

Abstract

We consider weighted $L^p(w)$ boundedness ($1<p<\infty$ and $w$ a Muckenhoupt $A_p$ weight) of the Calderón commutator ${\mathcal{C}}_{\hat{I}}\text{c◯}$ associated with rough homogeneous kernel, under the condition $\hat{I}\text{c◯}\in L^q({\mathbb{S}}^{n-1})$ for $q_0<q\le \infty$ with $q_0$ a fixed constant depending on $w$. Comparing to the previous related known results (assuming $\hat{I}\text{c◯}\in L^{\infty }({\mathbb{S}}^{n-1})$), our result for $\hat{I}\text{c◯}\in L^q({\mathbb{S}}^{n-1})$ with $q$ in the range $(q_0,\infty )$ is new. We also obtain a quantitative weighted bound for this ${\mathcal{C}}_{\hat{I}}\text{c◯}$ on $L^p(w)$, which is the best known quantitative result for this class of operators.

weaken the assumption on the kernel