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

TL;DR

This paper characterizes the Lebesgue points of the gradient of Riesz potentials for finite measures, linking the existence of principal values to measure decay conditions, and explores Lebesgue points for Cauchy integrals on Cantor sets.

Contribution

It provides a precise criterion for Lebesgue points of Riesz potential gradients, connecting measure decay with principal value existence, and investigates Lebesgue points for Cauchy integrals on fractal sets.

Findings

Lebesgue points characterized by measure decay condition

Principal value existence equivalent to Lebesgue point condition

Analysis of Lebesgue points for Cauchy integrals on Cantor sets

Abstract

Given a finite nonnegative Borel measure $m$ in $\mathbb{R}^{d}$, we identify the Lebesgue set $\mathcal{L}(V_{s}) \subset \mathbb{R}^{d}$ of the vector-valued function $$V_{s}(x) = \int_{\mathbb{R}^{d}}\frac{x - y}{|x - y|^{s + 1}} \mathrm{d}m(y), $$ for any order $0 < s < d$. We prove that $a \in \mathcal{L}(V_{s})$ if and only if the integral above has a principal value at $a$ and $$\lim_{r \to 0}{\frac{m(B_{r}(a))}{r^{s}}} = 0.$$ In that case, the precise representative of $V_{s}$ at $a$ 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.