Lipshitzian Vector Fields, Upper Gradients And Distributional Derivatives
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914766512979968 |
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| author | Venturini, Sergio |
| author_facet | Venturini, Sergio |
| contents | We prove that given a locally integrable function $f$ on an open set of an Euclidean space the distributional derivative $Xf$ with respect to a locally Lipshitzian vector field $X$ is locally integrable if, and only if, the function $f$ admits a locally integrable upper gradient along the vector field $X$; in this case $Xf$ coincides with the Lie derivative $L_X f$ and $|Xf|$ is the least upper gradient of the function $f$. Applications to systems of locally Lipshitzian vector fields are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10422 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lipshitzian Vector Fields, Upper Gradients And Distributional Derivatives Venturini, Sergio Metric Geometry 53C23 (Primary) 46E36 (Secondary) We prove that given a locally integrable function $f$ on an open set of an Euclidean space the distributional derivative $Xf$ with respect to a locally Lipshitzian vector field $X$ is locally integrable if, and only if, the function $f$ admits a locally integrable upper gradient along the vector field $X$; in this case $Xf$ coincides with the Lie derivative $L_X f$ and $|Xf|$ is the least upper gradient of the function $f$. Applications to systems of locally Lipshitzian vector fields are given. |
| title | Lipshitzian Vector Fields, Upper Gradients And Distributional Derivatives |
| topic | Metric Geometry 53C23 (Primary) 46E36 (Secondary) |
| url | https://arxiv.org/abs/2404.10422 |