A DEFINABLE p-ADIC ANALOGUE OF KIRSZBRAUN'S THEOREM ON EXTENSIONS OF LIPSCHITZ MAPS

作者：Cluckers Raf^*; Martin Florent

来源：Journal of the Institute of Mathematics of Jussieu, 2018, 17(1): 39-57.

DOI：10.1017/S1474748015000390

• p-adic semi-algebraic functions
• p-adic subanalytic functions
• Lipschitz continuous functions
• p-adic cell decomposition
• definable retractions

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.

• 出版日期2018-2