H\"older and Lipschitz continuity of functions definable over Henselian rank one valued fields

TL;DR

This paper proves that definable functions over Henselian rank one valued fields are H"older continuous and, under local Lipschitz conditions, are globally Lipschitz, advancing understanding of their regularity properties.

Contribution

It establishes H"older and Lipschitz continuity for definable functions over Henselian rank one valued fields, providing new regularity results in this setting.

Findings

Definable functions are H"older continuous with some exponent.

Locally Lipschitz functions are globally Lipschitz with possibly larger constants.

Results apply to functions over Henselian rank one valued fields.

Abstract

Consider a Henselian rank one valued field $K$ of equicharacteristic zero with the three-sorted language $\mathcal{L}$ of Denef--Pas. Let $f: A \to K$ be a continuous $\mathcal{L}$-definable (with parameters) function on a closed bounded subset $A \subset K^{n}$. The main purpose is to prove that then $f$ is H\"older continuous with some exponent $s\geq 0$ and constant $c \geq 0$, a fortiori, $f$ is uniformly continuous. Further, if $f$ is locally Lipschitz continuous with a constant $c$, then $f$ is (globally) Lipschitz continuous with possibly some larger constant $d$. Also stated are some problems concerning continuous and Lipschitz continuous functions definable over Henselian valued fields.