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arXiv · 2504.16657

A new characterization of Sobolev spaces on Lipschitz differentiability spaces

Bang-Xian Han·Zhe-Feng Xu·Zhuo-Nan Zhu
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Abstract

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 Poincar\'e 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.

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BibTeXRIS

Bang-Xian Han, Zhe-Feng Xu, Zhuo-Nan Zhu. 2025-04-23. A new characterization of Sobolev spaces on Lipschitz differentiability spaces. https://arxiv.org/abs/2504.16657

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