Semialgebraic Calderon-Zygmund theorem on regularization of the distance function
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| Format: | Preprint |
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2024
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| _version_ | 1866911845658394624 |
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| author | Kocel-Cynk, Beata Pawłucki, Wiesław Valette, Anna |
| author_facet | Kocel-Cynk, Beata Pawłucki, Wiesław Valette, Anna |
| contents | We prove that, for any closed semialgebraic subset $W$ of $\mathbb{R}^n$ and for any positive integer $p$, there exists a Nash function $f:\mathbb{R}^n\setminus W\longrightarrow (0, \infty)$ which is equivalent to the distance function from $W$ and at the same time it is $Λ_p$-regular in the sense that $|D^αf(x)|\leq C d(x, W)^{1- |α|}$, for each $x\in \mathbb{R}^n\setminus W$ and each $α\in \mathbb{N}^n$ such that $1\leq |α|\leq p$, where $C$ is a positive constant. In particular, $f$ is Lipschitz. Some applications of this result are given. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_03135 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semialgebraic Calderon-Zygmund theorem on regularization of the distance function Kocel-Cynk, Beata Pawłucki, Wiesław Valette, Anna Classical Analysis and ODEs Algebraic Geometry 14P20, 57R35, 14P10, 32B20 We prove that, for any closed semialgebraic subset $W$ of $\mathbb{R}^n$ and for any positive integer $p$, there exists a Nash function $f:\mathbb{R}^n\setminus W\longrightarrow (0, \infty)$ which is equivalent to the distance function from $W$ and at the same time it is $Λ_p$-regular in the sense that $|D^αf(x)|\leq C d(x, W)^{1- |α|}$, for each $x\in \mathbb{R}^n\setminus W$ and each $α\in \mathbb{N}^n$ such that $1\leq |α|\leq p$, where $C$ is a positive constant. In particular, $f$ is Lipschitz. Some applications of this result are given. |
| title | Semialgebraic Calderon-Zygmund theorem on regularization of the distance function |
| topic | Classical Analysis and ODEs Algebraic Geometry 14P20, 57R35, 14P10, 32B20 |
| url | https://arxiv.org/abs/2403.03135 |