Lebesgue points via the Poincar\'e inequality

TL;DR

This paper demonstrates that in certain metric measure spaces satisfying specific conditions, functions with a Poincaré inequality have Lebesgue points almost everywhere, extending classical results to more general spaces.

Contribution

It establishes the existence of Lebesgue points for functions satisfying a Poincaré inequality in Q-doubling spaces with a chain condition, generalizing classical Euclidean results.

Findings

Lebesgue points exist for functions with a Poincaré inequality in Q-doubling spaces.

The result applies to functions in Sobolev spaces on complete metric measure spaces.

Lebesgue points are shown to exist almost everywhere with respect to a specific Hausdorff measure.

Abstract

In this article, we show that in a $Q$-doubling space $(X,d,\mu),$ $Q>1,$ which satisfies a chain condition, if we have a $Q$-Poincar\'e inequality for a pair of functions $(u,g)$ where $g\in L^Q(X),$ then $u$ has Lebesgue points $H^h$-a.e. for $h(t)=\log^{1-Q-\epsilon}(1/t).$ We also discuss how the existence of Lebesgue points follows for $u\in W^{1,Q}(X)$ where $(X,d,\mu)$ is a complete $Q$-doubling space supporting a $Q$-Poincar\'e inequality for $Q>1.$