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Preserving Besov (fractional Sobolev) energies under sphericalization and flattening

Anders Björn, Jana Björn, Riikka Korte, Sari Rogovin, Timo Takala

TL;DR

This work develops a general, measure-theoretic framework for converting unbounded metric spaces into bounded ones (and vice versa) via sphericalization and flattening, while preserving key analytic structure. By introducing a metric deformation with a density function $\rho$ and a weighted measure $\hat{\nu}$, the authors prove that the transformed spaces remain doubling and that Besov energies are comparable when $\sigma = p\theta$, enabling nonlocal energy analysis on fractal-like sets to be transferred between bounded and unbounded settings. A central achievement is establishing the duality between sphericalization and flattening: each operation is invertible up to biLipschitz equivalence of the metrics and the measures, and inversions are included in the framework. The results provide a robust toolset for studying Besov-type energies and nonlocal PDEs on general metric measure spaces, including totally disconnected fractals, by transporting problems to more tractable bounded domains without loss of essential structure.

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.

Preserving Besov (fractional Sobolev) energies under sphericalization and flattening

TL;DR

This work develops a general, measure-theoretic framework for converting unbounded metric spaces into bounded ones (and vice versa) via sphericalization and flattening, while preserving key analytic structure. By introducing a metric deformation with a density function and a weighted measure , the authors prove that the transformed spaces remain doubling and that Besov energies are comparable when , enabling nonlocal energy analysis on fractal-like sets to be transferred between bounded and unbounded settings. A central achievement is establishing the duality between sphericalization and flattening: each operation is invertible up to biLipschitz equivalence of the metrics and the measures, and inversions are included in the framework. The results provide a robust toolset for studying Besov-type energies and nonlocal PDEs on general metric measure spaces, including totally disconnected fractals, by transporting problems to more tractable bounded domains without loss of essential structure.

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.
Paper Structure (13 sections, 29 theorems, 61 equations)

This paper contains 13 sections, 29 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sphericalizations, flattenings and inversions
  4. Uniform perfectness at a point
  5. Doubling measures
  6. The Besov (fractional Sobolev) energy
  7. Deformation of the metric
  8. Sphericalization and flattening
  9. The doubling property of the deformed measure
  10. Preservation of the Besov energy
  11. Duality of sphericalization and flattening
  12. The flattening of a sphericalized space
  13. The sphericalization of a flattened space

Key Result

Theorem 1.1

Assume that $(Z,d)$ is a complete metric space that is uniformly perfect at a base point $b\in Z$ for radii $r\ge m_0$ and that the measure $\nu$ is doubling. Let the deformed space $(\widehat{Z},\hat{d},{\hat{\nu}})$ be defined using eq-def-dhat-intro--eq-nuhat-intro below.

Theorems & Definitions (31)

  • Theorem 1.1
  • Proposition 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Lemma 3.5
  • Proposition 3.6
  • ...and 21 more