Wiener-Lebesgue point property for Sobolev Functions on Metric Spaces

TL;DR

This paper proves a Wiener-type integral condition for Sobolev functions on metric spaces, strengthening the Lebesgue point property with minimal capacity increase for non-Lebesgue points.

Contribution

It introduces a Wiener-type integral condition for Sobolev functions on metric measure spaces, extending the Lebesgue point property.

Findings

Established a Wiener-type integral condition for Sobolev functions.

Demonstrated the condition is stronger than the Lebesgue point property.

Showed the increase in capacity of non-Lebesgue points is marginal.

Abstract

We establish a Wiener-type integral condition for first-order Sobolev functions defined on a complete, doubling metric measure space supporting a Poincar\'e inequality. It is stronger than the Lebesgue point property, except for a marginal increase in the capacity of the set of non-Lebesgue points.