Doubling measures and Poincar\'e inequalities for sphericalizations of metric spaces

TL;DR

This paper studies how sphericalization of metric spaces preserves measures, Poincaré inequalities, and p-harmonic functions, leading to new boundary regularity results for unbounded domains, even in classical Euclidean spaces.

Contribution

It extends the preservation of p-harmonic functions and Poincaré inequalities under sphericalization to spaces with only doubling measures, broadening previous results that required Ahlfors regularity.

Findings

Preservation of p-harmonic functions under sphericalization.

New boundary regularity results at infinity for p-harmonic functions.

Applicability to unweighted Euclidean spaces for p ≠ 2.

Abstract

The identification between the complex plane and the Riemann sphere preserves holomorphic and harmonic functions and is a classical tool. In this paper we consider a similar mapping from an unbounded metric space $X$ to a bounded space and show how it preserves $p$-harmonic functions and Poincar\'e inequalities. When $X$ is Ahlfors regular, this was shown in our earlier paper (J. Math. Anal. Appl. 474 (2019), 852-875). Here we only require the much weaker (and more natural) doubling property of the measure. Furthermore, we consider a broader class of transformed measures. The sphericalization is then applied to obtain new results for the Dirichlet boundary value problem in unbounded sets and for boundary regularity at infinity for $p$-harmonic functions. Some of these results are new also for unweighted $\mathbf{R}^n$, $n \ge 2$ and $p\ne2$.