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arXiv · 2402.11174

Maz'ya-Shaposhnikova meet Bishop-Gromov

Bang-Xian Han·Andrea Pinamonti·Zhefeng Xu·Kilian Zambanini
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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.

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Bang-Xian Han, Andrea Pinamonti, Zhefeng Xu, Kilian Zambanini. 2024-02-17. Maz'ya-Shaposhnikova meet Bishop-Gromov. https://arxiv.org/abs/2402.11174

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