A refined Lusin type theorem for gradients
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915031116939264 |
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| author | De Masi, Luigi Marchese, Andrea |
| author_facet | De Masi, Luigi Marchese, Andrea |
| contents | We prove a refined version of the celebrated Lusin type theorem for gradients by Alberti, stating that any Borel vector field $f$ coincides with the gradient of a $C^1$ function $g$, outside a set $E$ of arbitrarily small Lebesgue measure. We replace the Lebesgue measure with any Radon measure $μ$, and we obtain that the estimate on the $L^p$ norm of $Dg$ does not depend on $μ(E)$, if the value of $f$ is $μ$-a.e. orthogonal to the decomposability bundle of $μ$. We observe that our result implies the 1-dimensional version of the flat chain conjecture by Ambrosio and Kirchheim on the equivalence between metric currents and flat chains with finite mass in $\mathbb{R}^n$ and we state a suitable generalization for $k$-forms, which would imply the validity of the conjecture in full generality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15012 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A refined Lusin type theorem for gradients De Masi, Luigi Marchese, Andrea Analysis of PDEs Metric Geometry 49Q15, 49Q20 We prove a refined version of the celebrated Lusin type theorem for gradients by Alberti, stating that any Borel vector field $f$ coincides with the gradient of a $C^1$ function $g$, outside a set $E$ of arbitrarily small Lebesgue measure. We replace the Lebesgue measure with any Radon measure $μ$, and we obtain that the estimate on the $L^p$ norm of $Dg$ does not depend on $μ(E)$, if the value of $f$ is $μ$-a.e. orthogonal to the decomposability bundle of $μ$. We observe that our result implies the 1-dimensional version of the flat chain conjecture by Ambrosio and Kirchheim on the equivalence between metric currents and flat chains with finite mass in $\mathbb{R}^n$ and we state a suitable generalization for $k$-forms, which would imply the validity of the conjecture in full generality. |
| title | A refined Lusin type theorem for gradients |
| topic | Analysis of PDEs Metric Geometry 49Q15, 49Q20 |
| url | https://arxiv.org/abs/2411.15012 |