Sparse domination of Calderón-Zygmund operators by mean oscillations

Abstract

We show that if $T$ is a Dini-continuous Calderón--Zygmund operator satisfying $T(1)=0$, then the usual sparse domination for $T$ can be sharpened by replacing local averages with local mean oscillations. This extends a result of Benea and Bernicot for smoother kernels to the more general Dini-continuous setting. As an application, we characterize the Calderón--Zygmund operators for which a pointwise Sobolev-type inequality holds: this is the case if and only if $T(1)\in L^{\infty }$. This answers a recent question of Hoang, Moen and Pérez.

Updated version. Sparse domination by mean oscillations was previously obtained by C. Benea and F. Bernicot under a stronger smoothness assumption on the kernel. The introduction has been revised accordingly