Abstract
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 \(\Lambda _p\)-regular in the sense that \(|D^\alpha f(x)|\le C d(x, W)^{1- |\alpha |}\), for each \(x\in {\mathbb {R}}^n{\setminus } W\) and each \(\alpha \in {\mathbb {N}}^n\) such that \(1\le |\alpha |\le p\), where C is a positive constant. In particular, f is Lipschitz. Some applications of this result are given.
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Notes
By [10], permutations of coordinates \(x_1,\dots , x_n\) suffice.
\(x_j\) was omitted as obviously \(\Lambda ^1_p(W)\)-regular.
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The authors thank the anonymous referees whose comments improved the first version of the paper.
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Anna Valette was partially supported by Narodowe Centrum Nauki (Poland); Grant no. 2021/43/B/ST1/02359.
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Kocel-Cynk, B., Pawłucki, W. & Valette, A. Semialgebraic Calderón-Zygmund theorem on regularization of the distance function. Math. Ann. (2024). https://doi.org/10.1007/s00208-023-02795-4
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DOI: https://doi.org/10.1007/s00208-023-02795-4