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arXiv · 2407.13051

New Characterizations of First Order Sobolev Spaces

Przemysław Górka·Kacper Kurowski
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Abstract

We provide new characterizations of Sobolev spaces that are true under some mild conditions. We study modified first order Sobolev spaces on metric measure spaces: $\mathrm{TC}$-Newtonian space, $\hat{\mathrm{TC}}$-Newtonian space, and Gigli-like space. We prove that if the measure is Borel regular and $\sigma$-finite, then the modified $\mathrm{TC}$-Newtonian space is equivalent to the Haj{\l}asz-Sobolev space. Moreover, if additionally the measure is doubling then all modified spaces are equivalent to the Haj{\l}asz-Sobolev space.

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Przemysław Górka, Kacper Kurowski. 2024-07-17. New Characterizations of First Order Sobolev Spaces. https://arxiv.org/abs/2407.13051

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