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On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: asymptotic volume ratio, volume entropy and rigidity

Bang-Xian Han, Andrea Pinamonti

Abstract

We study the asymptotic behaviour of suitably defined seminorms in general metric measure spaces. As a particular case we provide new and shorter proofs of the Maz'ya-Shaposhnikova's theorem on the asymptotic behaviour of the fractional Sobolev $s$-seminorm, in the setting of metric measure spaces and with general mollifiers, as well as of the Ludwig's result on finite dimensional Banach spaces. Our result also provides new spaces satisfying an asymptotic formula and it also builds a link between the asymptotic formula for functions and the asymptotic volume ratio of a metric measure space. In addition, we prove two related rigidity results for metric measure spaces with synthetic Ricci curvature bound which are new even in the smooth setting.

On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: asymptotic volume ratio, volume entropy and rigidity

Abstract

We study the asymptotic behaviour of suitably defined seminorms in general metric measure spaces. As a particular case we provide new and shorter proofs of the Maz'ya-Shaposhnikova's theorem on the asymptotic behaviour of the fractional Sobolev -seminorm, in the setting of metric measure spaces and with general mollifiers, as well as of the Ludwig's result on finite dimensional Banach spaces. Our result also provides new spaces satisfying an asymptotic formula and it also builds a link between the asymptotic formula for functions and the asymptotic volume ratio of a metric measure space. In addition, we prove two related rigidity results for metric measure spaces with synthetic Ricci curvature bound which are new even in the smooth setting.

Paper Structure

This paper contains 5 sections, 8 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. General theory
  4. Applications and Examples
  5. Rigidity results

Key Result

Theorem 2.2

Given $p>1$ and a metric measure space $(X,{\mathrm d},\mathfrak m)$. Let $(\rho_n)_{n\in \mathbb{N}}$ be a family of mollifiers. For any $u\in \cup_{n\in \mathbb{N}} W(n, p)$, there exists the limit where $L$ is the constant given in eq1:assumption.

Theorems & Definitions (18)

  • Theorem 2.2: Generalized Maz'ya-Shaposhnikova's formula
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Theorem 2.7
  • Definition 2.8
  • ...and 8 more