Sharp conditions for preserving uniformity, doubling measure and Poincaré inequality under sphericalization

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.