Quantitative weighted bounds for Calder\'{o}n commutator with rough kernel

TL;DR

This paper establishes new weighted boundedness results for the Calderón commutator with rough kernels in $L^p(w)$ spaces, extending previous results to less regular kernels and providing the best quantitative bounds known.

Contribution

It introduces the first weighted $L^p(w)$ bounds for Calderón commutators with kernels in $L^q$ for $q$ less than infinity, improving upon prior results that required bounded kernels.

Findings

Proved weighted boundedness for $ abla imes abla$ Calderón commutator with rough kernels.

Derived the sharpest quantitative bounds for these operators.

Extended the class of kernels for which weighted bounds are known.

Abstract

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