An endpoint estimate for the commutators of singular integral operators with rough kernels

TL;DR

This paper establishes an endpoint estimate for the commutators of singular integral operators with rough kernels, showing they satisfy an $L ext{log}L$ type bound under certain conditions.

Contribution

It proves an endpoint estimate for commutators of singular integrals with rough kernels, extending the understanding of their boundedness properties.

Findings

Commutators satisfy an $L ext{log}L$ endpoint estimate.

The result applies to kernels with $ ext{L}( ext{log} ext{L})^2$ regularity.

The work advances the theory of singular integrals with rough kernels.

Abstract

Let $\Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{d-1}$, $T_{\Omega}$ be the homogeneous singular integral operator with kernel $\frac{\Omega(x)}{|x|^d}$ and $T_{\Omega,\,b}$ be the commutator of $T_{\Omega}$ with symbol $b$. In this paper, we prove that if $\Omega\in L(\log L)^2(S^{d-1})$, then for $b\in {\rm BMO}(\mathbb{R}^d)$, $T_{\Omega,\,b}$ satisfies an endpoint estimate of $L\log L$ type.