Sharp conditions for preserving uniformity, doubling measure and Poincar\'e inequality under sphericalization

TL;DR

This paper establishes precise conditions under which sphericalization preserves key geometric and measure-theoretic properties like uniformity, doubling measure, and Poincaré inequality in metric measure spaces.

Contribution

It provides sharp criteria for the deforming density function ensuring the preservation of these properties during sphericalization.

Findings

Identifies sharp conditions for property preservation

Demonstrates the necessity of these conditions with examples

Extends understanding of metric measure space transformations

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 Poincar\'e inequality. We also provide examples that demonstrate the sharpness of our conditions.