Uniform extension of definable Cm,ω-Whitneyjets

We show that definable Whitney jets of class C m, , where m is a nonnegative integer and is a modulus of continuity, are the restrictions of definable C m, -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, extends to a definable bounded family of C m,functions.We also discuss a uniform C m -version and how the extension depends on the modulus of continuity.We prove a C m, -version of this result.Theorem 1.2.Let 0 m p be integers.Let be a modulus of continuity.Let E n be a definable closed set.Any definable Whitney jet of class C m, on E extends to a definable C m, -function on n which is of class C p outside E.By a modulus of continuity we always mean a positive, continuous, increasing, and concave function : (0, ) (0, ) such that (t) 0 as t 0. We say that is a modulus of continuity for a function f : S , defined on a subset S n , if there exists a constant C > 0 such that (1-1)| f (x) -f (y)| C (|x -y|) for all x, y S.