Maz'ya-Shaposhnikova meet Bishop-Gromov
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911778878783488 |
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| author | Han, Bang-Xian Pinamonti, Andrea Xu, Zhefeng Zambanini, Kilian |
| author_facet | Han, Bang-Xian Pinamonti, Andrea Xu, Zhefeng Zambanini, Kilian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11174 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maz'ya-Shaposhnikova meet Bishop-Gromov Han, Bang-Xian Pinamonti, Andrea Xu, Zhefeng Zambanini, Kilian Metric Geometry Functional Analysis 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. |
| title | Maz'ya-Shaposhnikova meet Bishop-Gromov |
| topic | Metric Geometry Functional Analysis |
| url | https://arxiv.org/abs/2402.11174 |