A new characterization of Sobolev spaces on Lipschitz differentiability spaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914465300086784 |
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| author | Han, Bang-Xian Xu, Zhe-Feng Zhu, Zhuo-Nan |
| author_facet | Han, Bang-Xian Xu, Zhe-Feng Zhu, Zhuo-Nan |
| contents | Numerous characterizations of Sobolev norms via the asymptotic behavior of non-local functionals have been established over the past decades; however, their validity beyond the PI framework remains poorly understood. We establish such a characterization on Lipschitz differentiability spaces without assuming either the doubling condition or a Poincaré inequality, by proving sharp two-sided Brezis--Van Schaftingen--Yung type asymptotic formulas. We also construct sharp counterexamples revealing the necessity of our assumptions, and provide several examples which are of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16657 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new characterization of Sobolev spaces on Lipschitz differentiability spaces Han, Bang-Xian Xu, Zhe-Feng Zhu, Zhuo-Nan Functional Analysis Metric Geometry Numerous characterizations of Sobolev norms via the asymptotic behavior of non-local functionals have been established over the past decades; however, their validity beyond the PI framework remains poorly understood. We establish such a characterization on Lipschitz differentiability spaces without assuming either the doubling condition or a Poincaré inequality, by proving sharp two-sided Brezis--Van Schaftingen--Yung type asymptotic formulas. We also construct sharp counterexamples revealing the necessity of our assumptions, and provide several examples which are of independent interest. |
| title | A new characterization of Sobolev spaces on Lipschitz differentiability spaces |
| topic | Functional Analysis Metric Geometry |
| url | https://arxiv.org/abs/2504.16657 |