# The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications

## TL;DR

This paper investigates boundary regularity for p-harmonic functions in metric spaces, establishing the Kellogg property for various compactifications and providing examples where it fails, extending known Euclidean results.

## Contribution

It extends the Kellogg property to the Mazurkiewicz boundary and other compactifications in metric spaces, including new results in Euclidean spaces.

## Key findings

- Kellogg property holds for many compactifications in metric spaces
- Examples are provided where the Kellogg property fails
- Results are new even in unweighted Euclidean spaces

## Abstract

In this paper boundary regularity for p-harmonic functions is studied with respect to the Mazurkiewicz boundary and other compactifications. In particular, the Kellogg property (which says that the set of irregular boundary points has capacity zero) is obtained for a large class of compactifications, but also two examples when it fails are given. This study is done for complete metric spaces equipped with doubling measures supporting a p-Poincar\'e inequality, but the results are new also in unweighted Euclidean spaces.