Whitney's extension problem for multivariate C-1,C-w-functions

We prove that the trace of the space C-1,C-w (R-n) to an arbitrary closed s ubset X subset of R-n is characterized by the following "finiteness" proper ty. A function f : X --> R belongs to the trace space if and only if the re striction f/(Y) to an arbitrary subset Y subset of X consisting of at most 3 .2(n-1) can be extended to a function f(Y) is an element of C-1,C-w (R-n) such that sup {\\f(Y)\\C-1,C-w : Y subset of X, card Y less than or equal to 3 . 2(n- 1)} < <infinity>. The constant 3 . 2(n-1) is sharp. The proof is based on a Lipschitz selection result which is interesting in its own right.