On the $L^p$-differentiability of certain classes of functions

Abstract

We prove the $L^p$-differentiability at almost every point for convolution products on ${\mathbb{R}}^d$ of the form $K\ast \mu$, where $\mu$ is bounded measure and $K$ is a homogeneous kernel of degree $1-d$. From this result we derive the $L^p$-differentiability for vector fields on ${\mathbb{R}}^d$ whose curl and divergence are measures, and also for vector fields with bounded deformation.