Semialgebraic Calderón-Zygmund theorem on regularization of the distance function

We prove that, for any closed semialgebraic subset W of R n and for any positive integer p , there exists a Nash function f : R n \ W ⟶ ( 0 , ∞ ) which is equivalent to the distance function from W and at the same time it is Λ p -regular in the sense that | D α f ( x ) | ≤ C d ( x , W ) 1 - | α | ,...

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Bibliographic Details
Published in:Mathematische annalen 2024, Vol.390 (2), p.1863-1883
Main Authors: Kocel-Cynk, Beata, Pawłucki, Wiesław, Valette, Anna
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Regularization
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Summary:We prove that, for any closed semialgebraic subset W of R n and for any positive integer p , there exists a Nash function f : R n \ W ⟶ ( 0 , ∞ ) which is equivalent to the distance function from W and at the same time it is Λ p -regular in the sense that | D α f ( x ) | ≤ C d ( x , W ) 1 - | α | , for each x ∈ R n \ W and each α ∈ N n such that 1 ≤ | α | ≤ p , where C is a positive constant. In particular, f is Lipschitz. Some applications of this result are given.
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-023-02795-4