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LIpSCHITZ CONTINUITY pROpERTIES FOR p-adic SEMI-ALGEBRAIC AND subanalytic functions

GEOMETRIC AND FUNCTIONAL ANALYSIS 2010年第1期20卷 68-87页

作者： Cluckers, Raf Comte, Georges Loeser, Francois Univ Lille 1 Lab Painleve CNRS UMR 8524 F-59655 Villeneuve Dascq France Katholieke Univ Leuven Dept Math B-3001 Leuven Belgium Univ Nice Sophia Antipolis Lab JA Dieudonne F-06108 Nice 02 France Ecole Normale Super Dept Math & Applicat CNRS UMR 8553 F-75230 Paris 05 France

We prove that a (globally) subanalytic function f : X subset of Q(p)(n) -> Q(p) which is locally Lipschitz continuous with some constant C is piecewise (globally on each piece) Lipschitz continuous with possibly some other constant, where the pieces can be taken to be subanalytic. We also prove the analogous result for a subanalytic family of functions f(y) : X(y) subset of Q(p)(n) -> Q(p) depending on p-adic parameters. The statements also hold in a semi-algebraic set-up and also in a finite field extension of Q(p). These results are p-adic analogues of results of K. Kurdyka over the real numbers. To encompass the total disconnectedness of p-adic fields, we need to introduce new methods adapted to the p-adic situation.

关键词： p-adic semi-algebraic functions p-adic subanalytic functions Lipschitz continuous functions cell decomposition

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Approximations and Lipschitz continuity in p-adic semi-algebraic and subanalytic geometry

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SELECTA MATHEMATICA-NEW SERIES 2012年第4期18卷 825-837页

作者： Cluckers, Raf Halupczok, Immanuel Univ Lille 1 Lab Painleve CNRS UMR 8524 F-59655 Villeneuve Dascq France Katholieke Univ Leuven Dept Math B-3001 Louvain Belgium Univ Munster Inst Math Log & Grundlagenforsch D-48149 Munster Germany

It is known that a p-adic, locally Lipschitz continuous semi-algebraic function, is piecewise Lipschitz continuous, where finitely many pieces suffice and the pieces can be taken semi-algebraic. We prove that if the function has locally Lipschitz constant 1, then it is also piecewise Lipschitz continuous with the same Lipschitz constant 1 (again, with finitely many pieces). We do this by proving the following fine preparation results for p-adic semi-algebraic functions in one variable. Any such function can be well approximated by a monomial with fractional exponent such that moreover the derivative of the monomial is an approximation of the derivative of the function. We also prove these results in parameterized versions and in the subanalytic setting.

关键词： p-adic semi-algebraic functions p-adic subanalytic functions Lipschitz continuous functions p-adic cell decomposition

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A DEFINABLE p-adic ANALOGUE OF KIRSZBRAUN'S THEOREM ON EXTENSIONS OF LIpSCHITZ MApS

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JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU 2018年第1期17卷 39-57页

作者： Cluckers, Raf Martin, Florent Univ Lille Lab Painleve CNRS UMR 8524 F-59655 Villeneuve Dascq France Katholieke Univ Leuven Dept Math Celestijnenlaan 200B B-3001 Leuven Belgium Univ Regensburg Fak Math D-93040 Regensburg Germany

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.

关键词： p-adic semi-algebraic functions p-adic subanalytic functions Lipschitz continuous functions p-adic cell decomposition definable retractions

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