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

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 $n \ge 2$.