Lipschitz continuity properties for p-adic semi-algebraic and subanalytic functions

TL;DR

This paper establishes that subanalytic p-adic functions that are locally Lipschitz continuous can be partitioned into subanalytic pieces where they are globally Lipschitz, extending real results to the p-adic context.

Contribution

It proves p-adic analogues of Kurdyka's results on Lipschitz continuity, adapting methods to the totally disconnected p-adic setting.

Findings

Locally Lipschitz p-adic functions are piecewise globally Lipschitz.

Results hold for semi-algebraic and p-adic parameter families.

Applicable to finite extensions of p-adic fields.

Abstract

We prove that a (globally) subanalytic p-adic function 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 subanalytic. We also prove the analogous result for a subanalytic family of functions depending on p-adic parameters. The statements also hold in a semi-algebraic set-up and also in finite extensions of the field of p-adic numbers. 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.