Sparse domination of Calderón-Zygmund operators by mean oscillations

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 $T(1)\in L^{\mathrm{\infty }}$. This answers a recent question of Hoang, Moen and Pérez.