Maz'ya-Shaposhnikova meet Bishop-Gromov
Metric Geometry
2024-02-20 v1 Functional Analysis
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.
Keywords
Cite
@article{arxiv.2402.11174,
title = {Maz'ya-Shaposhnikova meet Bishop-Gromov},
author = {Bang-Xian Han and Andrea Pinamonti and Zhefeng Xu and Kilian Zambanini},
journal= {arXiv preprint arXiv:2402.11174},
year = {2024}
}