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

TL;DR

This paper establishes $L^p$ boundedness for a class of Calderón commutators with rough kernels, under certain integrability and smoothness conditions, extending the understanding of their behavior in harmonic analysis.

Contribution

It proves $L^p$ boundedness for Calderón commutators with rough kernels under new logarithmic integrability conditions, broadening previous results in the field.

Findings

Boundedness on $L^p$ for $rac{2eta}{2eta-1}<p<2eta$

Conditions involving logarithmic integrability of the kernel

Extension of boundedness results to rough kernels with vanishing moments

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 Calder\'on commutator defined by $$T_{\Omega,\,a;k}f(x)={\rm p.\,v.}\int_{\mathbb{R}^d}\frac{\Omega(x-y)}{|x-y|^{d+k}}\big(a(x)-a(y)\big)^kf(y){d}y.$$ In this paper, the authors prove that if $$\sup_{\zeta\in S^{d-1}}\int_{S^{d-1}}|\Omega(\theta)|\log ^{\beta} \big(\frac{1}{|\theta\cdot\zeta|}\big)d\theta<\infty,$$ with $\beta\in(1,\,\infty]$, then for $\frac{2\beta}{2\beta-1}<p<2\beta$, $T_{\Omega,\,a;\,k}$ is bounded on $L^p(\mathbb{R}^d)$.