Whitney’s extension problem for multivariate 𝐶^{1,𝜔}-functions

Abstract

We prove that the trace of the space C 1 , ω ( R n ) C^{1,\omega }({\mathbb R}^n) to an arbitrary closed subset X ⊂ R n X\subset {\mathbb R}^n is characterized by the following “finiteness” property. A function f : X → R f:X\rightarrow {\mathbb R} belongs to the trace space if and only if the restriction f | Y f|_Y to an arbitrary subset Y ⊂ X Y\subset X consisting of at most 3 ⋅ 2 n − 1 3\cdot 2^{n-1} can be extended to a function f Y ∈ C 1 , ω ( R n ) f_Y\in C^{1,\omega }({\mathbb R}^n) such that \[ sup { ‖ f Y ‖ C 1 , ω :   Y ⊂ X ,   card ⁡ Y ≤ 3 ⋅ 2 n − 1 } > ∞ . \sup \{\|f_Y\|_{C^{1,\omega }}:~Y\subset X, ~\operatorname {card} Y\le 3\cdot 2^{n-1}\}>\infty . \] The constant 3 ⋅ 2 n − 1 3\cdot 2^{n-1} is sharp. The proof is based on a Lipschitz selection result which is interesting in its own right.