Cluckers, Raf and Martin, Florent (2018) A DEFINABLE p-ADIC ANALOGUE OF KIRSZBRAUN'S THEOREM ON EXTENSIONS OF LIPSCHITZ MAPS. JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU, 17 (1). pp. 39-57. ISSN 1474-7480, 1475-3030
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A direct application of Zorn's Lemma gives that every Lipschitz map f : X subset of Q(p)(n) -> Q(p)(l) has an extension to a Lipschitz map (f) over tilde : Q(p)(n) -> Q(p)(l). This is analogous, but more easy, to Kirszbraun's Theorem about the existence of Lipschitz extensions of Lipschitz maps S subset of R-n -> R-l. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun's Theorem. In the present paper, we prove in the p-adic context that (f) over tilde can be taken definable when f is definable, where definable means semi-algebraic or subanalytic (or, some intermediary notion). We proceed by proving the existence of definable, Lipschitz retractions of Q(p)(n) to the topological closure of X when X is definable.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | SETS; REAL; p-adic semi-algebraic functions; p-adic subanalytic functions; Lipschitz continuous functions; p-adic cell decomposition; definable retractions |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 20 Mar 2020 14:34 |
| Last Modified: | 20 Mar 2020 14:34 |
| URI: | https://pred.uni-regensburg.de/id/eprint/15165 |
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