The precise representative for the gradient of the Riesz potential of a finite measure

Abstract

Given a finite nonnegative Borel measure (Formula presented.) in (Formula presented.), we identify the Lebesgue set (Formula presented.) of the vector-valued function (Formula presented.) for any order (Formula presented.). We prove that (Formula presented.) if and only if the integral above has a principal value at (Formula presented.) and (Formula presented.) In that case, the precise representative of (Formula presented.) at (Formula presented.) coincides with the principal value of the integral. We also study the existence of Lebesgue points for the Cauchy integral of the intrinsic probability measure associated with planar Cantor sets, which leads to challenging new questions. © 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.