Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular metric spaces

TL;DR

This paper employs sphericalization to analyze p-harmonic functions on unbounded domains within Ahlfors regular metric spaces, focusing on boundary regularity at infinity and demonstrating non-subadditivity of p-harmonic measure in various settings.

Contribution

It introduces a sphericalization approach to study boundary regularity and p-harmonic measures, extending results to unbounded domains and multiple approach directions at infinity.

Findings

Boundary regularity at infinity depends on approach directions and complement massiveness.

p-harmonic measure is not subadditive on null sets for certain dimensions and weights.

Constructs examples in the unit ball illustrating non-subadditivity of p-harmonic measure.

Abstract

We use sphericalization to study the Dirichlet problem, Perron solutions and boundary regularity for p-harmonic functions on unbounded sets in Ahlfors regular metric spaces. Boundary regularity for the point at infinity is given special attention. In particular, we allow for several "approach directions" towards infinity and take into account the massiveness of their complements. In 2005, Llorente-Manfredi-Wu showed that the p-harmonic measure on the upper half space $R^n_+, n \ge 2$, is not subadditive on null sets when $p \neq 2$. Using their result and spherical inversion, we create similar bounded examples in the unit ball $B \subset R^n$ showing that the n-harmonic measure is not subadditive on null sets when $n \ge 3$, and neither are the p-harmonic measures in $B$ generated by certain weights depending on $p\neq 2$ and $n \ge 2$.