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On estimates of the Bourgain-Brezis-Mironescu type in the singular context

Roman D. Oleinik

TL;DR

This work advances BBM-type characterizations to the singular setting of metric measure spaces by broadening the mollifier class to include non-monotone families and by treating maps valued in arbitrary metric spaces. The authors leverage Cheeger energy, metric-valued Sobolev calculus, and a robust doubling–Poincaré framework to establish two-sided comparisons between the limiting nonlocal functional and the $p$-energy $E_p[f](O)$, up to a universal constant $C$. They introduce a flexible notion of $p$-admissible mollifiers, provide simplified, verifiable conditions, and verify admissibility for several explicit mollifier families, including nonmonotone examples. The results extend previous BBM-type analyses (LPZ22, O25) to broader mollifier classes and metric-valued targets, enabling nonlocal-to-local energy characterizations in highly non-smooth contexts and with potential applications to geometric analysis and nonlinear potential theory.

Abstract

We extend the characterization of the Bourgain-Brezis-Mironescu type for maps from certain metric measure spaces to arbitrary metric spaces to a broader class of mollifiers.

On estimates of the Bourgain-Brezis-Mironescu type in the singular context

TL;DR

This work advances BBM-type characterizations to the singular setting of metric measure spaces by broadening the mollifier class to include non-monotone families and by treating maps valued in arbitrary metric spaces. The authors leverage Cheeger energy, metric-valued Sobolev calculus, and a robust doubling–Poincaré framework to establish two-sided comparisons between the limiting nonlocal functional and the -energy , up to a universal constant . They introduce a flexible notion of -admissible mollifiers, provide simplified, verifiable conditions, and verify admissibility for several explicit mollifier families, including nonmonotone examples. The results extend previous BBM-type analyses (LPZ22, O25) to broader mollifier classes and metric-valued targets, enabling nonlocal-to-local energy characterizations in highly non-smooth contexts and with potential applications to geometric analysis and nonlinear potential theory.

Abstract

We extend the characterization of the Bourgain-Brezis-Mironescu type for maps from certain metric measure spaces to arbitrary metric spaces to a broader class of mollifiers.
Paper Structure (13 sections, 24 theorems, 96 equations)

This paper contains 13 sections, 24 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Results
  4. Preliminaries
  5. General notation
  6. Elements of Sobolev calculus
  7. Doubling and Poincaré conditions
  8. Requirements for mollifiers
  9. Proofs
  10. Lower bound
  11. Upper bound
  12. Simplified assumptions
  13. Admissible mollifiers

Key Result

Theorem 1.1

Let $(\mathsf{X}, \mathsf{d},\mathfrak{m})$ be a locally complete metric measure space. Let $p\in [1,+\infty)$. Suppose $(\mathsf{X}, \mathsf{d},\mathfrak{m})$ is a $p$-Poincaré space. Let $(\rho_{\delta})_{\delta\in (0,1)}$ be a $p$-admissible family. Then there exists a universal constant $C\in (0

Theorems & Definitions (69)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • ...and 59 more