Sharp conditions for preserving uniformity, doubling measure and Poincaré inequality under sphericalization
Riikka Korte, Sari Rogovin, Nageswari Shanmugalingam, Timo Takala
TL;DR
This work identifies sharp, verifiable conditions on a metric-density function $\rho$ that governs sphericalization, a transformation that maps unbounded metric measure spaces to bounded ones. By establishing precise density assumptions (A: $\rho$-doubling, B: upper bound on tails, and C: boundary-mass control) and assuming lower semicontinuity where needed, the authors prove that uniformity, doubling measure, and $p$-Poincaré inequalities are preserved under sphericalization, with quantitative control of the transformed objects $d_\rho$ and $\mu_\rho$. They also demonstrate sharpness of the conditions through explicit examples (notably the Euclidean half-plane) and develop a comprehensive framework showing how curvature-less, non-smooth spaces behave under this conformal deformation. The results enable transferring boundary-value and Dirichlet-type problems from unbounded domains to bounded, well-behaved spaces, facilitating variational and analytic methods in non-smooth metric measure spaces.
Abstract
We study sphericalization, which is a mapping that conformally deforms the metric and the measure of an unbounded metric measure space so that the deformed space is bounded. The goal of this paper is to study sharp conditions on the deforming density function under which the sphericalization preserves uniformity of the space, the doubling property of the measure and the support of a Poincaré inequality. We also provide examples that demonstrate the sharpness of our conditions.