Uniform extension of definable $C^{m,\omega}$-Whitney jets

Adam Parusiński and Armin Rainer

Vol. 330 (2024), No. 2, 317–353

DOI: 10.2140/pjm.2024.330.317

Abstract

We show that definable Whitney jets of class $C^{m,\omega }$, where $m$ is a nonnegative integer and $\omega$ is a modulus of continuity, are the restrictions of definable $C^{m,\omega }$-functions; “definable” refers to an arbitrary given o-minimal expansion of the real field. This is true in a uniform way: any definable bounded family of Whitney jets of class $C^{m,\omega }$ extends to a definable bounded family of $C^{m,\omega }$-functions. We also discuss a uniform $C^m$-version and how the extension depends on the modulus of continuity.

Keywords

o-minimal structures, Whitney extension theorem, $C^{m,\omega}$-extension of Whitney jets, uniform boundedness of the extension

Mathematical Subject Classification

Primary: 03C64, 14P10, 32B20

Secondary: 26B35, 26E25, 46E15

Milestones

Received: 2 November 2023

Revised: 25 June 2024

Accepted: 26 July 2024

Published: 13 September 2024