New Characterizations of First Order Sobolev Spaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929633648181248 |
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| author | Górka, Przemysław Kurowski, Kacper |
| author_facet | Górka, Przemysław Kurowski, Kacper |
| contents | 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 $σ$-finite, then the modified $\mathrm{TC}$-Newtonian space is equivalent to the Hajłasz-Sobolev space. Moreover, if additionally the measure is doubling then all modified spaces are equivalent to the Hajłasz-Sobolev space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13051 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New Characterizations of First Order Sobolev Spaces Górka, Przemysław Kurowski, Kacper Functional Analysis 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 $σ$-finite, then the modified $\mathrm{TC}$-Newtonian space is equivalent to the Hajłasz-Sobolev space. Moreover, if additionally the measure is doubling then all modified spaces are equivalent to the Hajłasz-Sobolev space. |
| title | New Characterizations of First Order Sobolev Spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2407.13051 |