Generalized Lebesgue points for Sobolev functions

TL;DR

This paper proves that functions in certain fractional Sobolev spaces on doubling metric measure spaces have generalized Lebesgue points outside a negligible set, extending classical Lebesgue differentiation results.

Contribution

It establishes the existence of generalized Lebesgue points for Sobolev functions with fractional order in a metric measure space setting, under minimal measure assumptions.

Findings

Generalized Lebesgue points exist outside a set of measure zero.

Results apply to functions in fractional Sobolev spaces with 0<s≤1 and 0<p<1.

The work extends classical differentiation theorems to more general metric spaces.

Abstract

In this article, we show that a function $f\in M^{s,p}(X),$ $0<s\leq 1,$ $0<p<1,$ where $X$ is a doubling metric measure space, has generalized Lebesgue points outside a set of $\mathcal{H}^h$-Hausdorff measure zero for a suitable gauge function $h.$