Whitney’s extension problem for multivariate $C^{1,\omega }$-functions

Abstract:

We prove that the trace of the space $C^{1,\omega }({\mathbb R}^n)$ to an arbitrary closed subset $X\subset {\mathbb R}^n$ is characterized by the following “finiteness” property. A function $f:X\rightarrow {\mathbb R}$ belongs to the trace space if and only if the restriction $f|_Y$ to an arbitrary subset $Y\subset X$ consisting of at most $3\cdot 2^{n-1}$ can be extended to a function $f_Y\in C^{1,\omega }({\mathbb R}^n)$ such that \[ \sup \{\|f_Y\|_{C^{1,\omega }}:~Y\subset X, ~\operatorname {card} Y\le 3\cdot 2^{n-1}\}<\infty . \] The constant $3\cdot 2^{n-1}$ is sharp.

The proof is based on a Lipschitz selection result which is interesting in its own right.