Skip to main content
×
Home
    • Aa
    • Aa

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

  • Raf Cluckers (a1) (a2) and Florent Martin (a3)
Abstract

A direct application of Zorn’s lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ has an extension to a Lipschitz map $\widetilde{f}:\mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ . This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell }$ . Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the $p$ -adic context that $\widetilde{f}$ 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 $\mathbb{Q}_{p}^{n}$ to the topological closure of $X$ when $X$ is definable.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

M. Aschenbrenner and A. Fischer , Definable versions of theorems by Kirszbraun and Helly, Proc. Lond. Math. Soc. (3) 102(3) (2011), 468502.

R. Cluckers , G. Comte and F. Loeser , Lipschitz continuity properties for p-adic semi-algebraic and subanalytic functions, Geom. Funct. Anal. 20(1) (2010), 6887.

R. Cluckers and I. Halupczok , Approximations and Lipschitz continuity in p-adic semi-algebraic and subanalytic geometry, Selecta Math. (N.S.) 18(4) (2012), 825837.

P. J. Cohen , Decision procedures for real and p-adic fields, Comm. Pure Appl. Math. 22 (1969), 131151.

J. Denef , The rationality of the Poincaré series associated to the p-adic points on a variety, Invent. Math. 77 (1984), 123.

J. Denef and L. van den Dries , p-adic and real subanalytic sets, Ann. of Math. (2) 128(1) (1988), 79138.

T. Kuijpers , Lipschitz extensions of definable p-adic functions, Math. Log. Q. 61(3) (2015), 151158.

A. Macintyre , On definable subsets of p-adic fields, J. Symbolic Logic 41 (1976), 605610.

E. J. McShane , Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837842.

M. Presburger , On the completeness of a certain system of arithmetic of whole numbers in which addition occurs as the only operation, Hist. Philos. Logic 12(2) (1991), 225233.

H. Whitney , Analytic extensions of functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 6389.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 7 *
Loading metrics...

Abstract views

Total abstract views: 58 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd September 2017. This data will be updated every 24 hours.