Korevaar-Schoen and heat kernel characterizations of Sobolev and BV spaces on local trees
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912376794644480 |
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| author | Baudoin, Fabrice Chen, Li Yang, Meng |
| author_facet | Baudoin, Fabrice Chen, Li Yang, Meng |
| contents | We study Sobolev and BV spaces on local trees which are metric spaces locally isometric to real trees. Such spaces are equipped with a Radon measure satisfying a locally uniform volume growth condition. Using the intrinsic geodesic structure, we define weak gradients and develop from it a coherent theory of Sobolev and BV spaces. We provide two main characterizations: one via Korevaar-Schoen-type energy functionals and another via the heat kernel associated with the natural Dirichlet form. Applications include interpolation results for Besov-Lipschitz spaces, critical exponents computations, and a Nash inequality. In globally tree-like settings we also establish $L^p$ gradient bounds for the heat semigroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10177 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Korevaar-Schoen and heat kernel characterizations of Sobolev and BV spaces on local trees Baudoin, Fabrice Chen, Li Yang, Meng Analysis of PDEs Functional Analysis Metric Geometry Probability We study Sobolev and BV spaces on local trees which are metric spaces locally isometric to real trees. Such spaces are equipped with a Radon measure satisfying a locally uniform volume growth condition. Using the intrinsic geodesic structure, we define weak gradients and develop from it a coherent theory of Sobolev and BV spaces. We provide two main characterizations: one via Korevaar-Schoen-type energy functionals and another via the heat kernel associated with the natural Dirichlet form. Applications include interpolation results for Besov-Lipschitz spaces, critical exponents computations, and a Nash inequality. In globally tree-like settings we also establish $L^p$ gradient bounds for the heat semigroup. |
| title | Korevaar-Schoen and heat kernel characterizations of Sobolev and BV spaces on local trees |
| topic | Analysis of PDEs Functional Analysis Metric Geometry Probability |
| url | https://arxiv.org/abs/2505.10177 |