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Maz'ya-Shaposhnikova meet Bishop-Gromov

Bang-Xian Han, Andrea Pinamonti, Zhefeng Xu, Kilian Zambanini

Abstract

We find a surprising link between Maz'ya-Shaposhnikova's well-known asymptotic formula concerning fractional Sobolev seminorms and the generalized Bishop-Gromov inequality. In the setting of abstract metric measure spaces we prove the validity of a large family of asymptotic formulas concerning non-local energies. Important examples which are covered by our approach are for instance Carnot groups, Riemannian manifolds with Ricci curvature bounded from below and non-collapsed RCD spaces. We also extend the classical Maz'ya-Shaposhnikova's formula on Euclidean spaces to a wider class of mollifiers.

Maz'ya-Shaposhnikova meet Bishop-Gromov

Abstract

We find a surprising link between Maz'ya-Shaposhnikova's well-known asymptotic formula concerning fractional Sobolev seminorms and the generalized Bishop-Gromov inequality. In the setting of abstract metric measure spaces we prove the validity of a large family of asymptotic formulas concerning non-local energies. Important examples which are covered by our approach are for instance Carnot groups, Riemannian manifolds with Ricci curvature bounded from below and non-collapsed RCD spaces. We also extend the classical Maz'ya-Shaposhnikova's formula on Euclidean spaces to a wider class of mollifiers.
Paper Structure (4 sections, 4 theorems, 72 equations)

This paper contains 4 sections, 4 theorems, 72 equations.

Table of Contents

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

Key Result

Theorem 2.2

Let $(X,{\mathrm d},\mathfrak m)$ be a metric measure space and let $(\rho_n)_{n\in \mathbb{N}}$ be mollifiers satisfying Assumption assumption1. Then, for any $u \in L^p$ such that $\mathcal{E}_{n_0}(u)<+\infty$ for a certain $n_0\in \mathbb{N}$, it holds

Theorems & Definitions (23)

  • Theorem 2.2: Generalized Maz'ya– Shaposhnikova's formula
  • proof
  • Remark 2.3
  • Definition 2.4: Generalized Bishop--Gromov inequality
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Example 2.9
  • Lemma 2.11
  • ...and 13 more