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

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.