Gradient regularity and first-order potential estimates for a class of nonlocal equations

Abstract

We consider nonlocal equations of order larger than one with measure data and prove gradient regularity in Sobolev and Hölder spaces as well as pointwise bounds of the gradient in terms of Riesz potentials, leading to fine regularity results in many commonly used function spaces. The kernel of the integral operators involves a Hölder dependence in the variables and is not assumed to be translation invariant.

56 pages. To appear in American Journal of Mathematics