Lebesgue points via the Poincaré inequality

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 $Q$-Poincaré 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 $Q$-Poincaré inequality for $Q>1.$

16 pages