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arXiv · 2508.09795

Doubling measures and Poincar\'e inequalities for sphericalizations of metric spaces, with applications to $p$-harmonic functions in unbounded domains

Anders Björn·Jana Björn·Xining Li
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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 ($\textit{J. Math. Anal. Appl. }\mathbf{474}$ (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 $p\ne2$.

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Anders Björn, Jana Björn, Xining Li. 2025-08-13. Doubling measures and Poincar\'e inequalities for sphericalizations of metric spaces, with applications to $p$-harmonic functions in unbounded domains. https://arxiv.org/abs/2508.09795

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