Submission date: 15 June 2023

Abstract:

Whitney's extension problem, i.e., how one can tell whether a function f : X → ℝ X ⊆ ℝ^n, is the restriction of a C^m-function on ℝ^n, was solved in full generality by Charles Fefferman in 2006. In this paper, we settle the C^{1,ω}-case of a related conjecture: given that f is semialgebraic and ω is a semialgebraic modulus of continuity, if f is the restriction of a C^{1,ω}-function then it is the restriction of a semialgebraic C^{1,ω}-function. We work in the more general setting of sets that are definable in an o-minimial expansion of the real field. An ingenious argument of Brudnyi and Shvartsman relates the existence of C^{1,ω}-extensions to the existence of Lipschitz selections of certain affine-set valued maps. We show that if a definable affine-set valued map has Lipschitz selections then it also has definable Lipschitz selections. In particular, we obtain a Lipschitz solution (more generally, ω-Hölder solution, for any definable modulus of continuity ω) of the definable Brenner-Epstein-Hochster-Kollár problem. In most of our results we have control over the respective (semi)norms.

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Full text arXiv 2306.09155: pdf, ps.