2023/06/15 by Parusiński, Adam, Rainer, Armin
#Algebraic Geometry (math.AG) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO)
paper · doi:10.48550/arxiv.2306.09155
Whitney's extension problem, i.e., how one can tell whether a function f : X → \mathbb R, X ⊆ \mathbb Rn, is the restriction of a Cm-function on \mathbb Rn, was solved in full generality by Charles Fefferman in 2006. In this paper, we settle the C1,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 C1,ω-function then it is the restriction of a semialgebraic C1,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 C1,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.