Doubling measures and Poincaré inequalities for sphericalizations of metric spaces, with applications to p-harmonic functions in unbounded domains

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 Poincaré inequalities. When X is Ahlfors regular, this was shown in our earlier paper (2019). 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 \(p\ne 2\).