Lipschitz Continuity Properties for p−Adic Semi-Algebraic and Subanalytic Functions

Abstract

We prove that a (globally) subanalytic function ${f : X \subset {\bf Q}^{n}_{p} \rightarrow {\bf Q}_{p}}$ 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 ${f_{y} : X_{y} \subset {\bf Q}^{n}_{p} \rightarrow {\bf Q}_{p}}$ 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.