Preserving Besov (fractional Sobolev) energies under sphericalization and flattening

TL;DR

This paper introduces a new sphericalization mapping for metric spaces that preserves Besov energies and doubling measures, applicable to fractal and disconnected sets, with implications for space equivalence under flattening and sphericalization.

Contribution

The paper develops a general sphericalization method that maintains key measure and energy properties, extending to fractal and disconnected metric spaces, and analyzes their biLipschitz equivalence after transformations.

Findings

The sphericalization preserves doubling measures and Besov energies.

Flattening and sphericalization compositions are biLipschitz equivalent to the original space.

The method applies to fractal and disconnected metric spaces.

Abstract

We introduce a new sphericalization mapping for metric spaces that is applicable in very general situations, including totally disconnected fractal type sets. For an unbounded complete metric space which is uniformly perfect at a base point for large radii and equipped with a doubling measure, we make a more specific construction based on the measure and equip it with a weighted measure. This mapping is then shown to preserve the doubling property of the measure and the Besov (fractional Sobolev) energy. The corresponding results for flattening of bounded complete metric spaces are also obtained. Finally, it is shown that for the composition of a sphericalization with a flattening, or vice versa, the obtained space is biLipschitz equivalent with the original space and the resulting measure is comparable to the original measure.