Fefferman's sharp form of Whitney's extension theorem

The problem addressed in this thesis is the following: suppose we are given a function f : E → R, where E is an arbitrary subset of R^n. How can we determine whether f extends to a C^{m−1,1} function on R^n? The version of this problem with C^m in place of C^{m−1,1} was originally answered by Whitney for functions defined on closed sets. To answer this question, Fefferman proved the following result. We only need to extend the function value f(xi) to a Taylor polynomial of order m-1 P_i at a fixed, finite number of points x_1, . . . , x_k, in order to obtain the existence of a C^{m-1,1} extension. To apply the standard Whitney’s extension theorem to our problem, we would need to extend f(x) to a Taylor polynomial P_x at every point x ∈ E. Note also that each P_i in the statement is allowed to depend on x_1, . . . , x_k, rather than on x_i alone. To prove the result, it is natural to look for functions F of bounded C^{m−1,1} norms on R^n, that agree with f on arbitrarily large finite subsets E_1 ⊂ E: we indeed arrive to a finite extension problem. With an easy argument, one can show that the problem reduces to an estimate of the order of magnitude (up to a constant depending only on m and n) of the infimum of the C^m-norms of smooth functions F : R^n → R that coincide with f on E_1. The tools used in the proof include Helly's Theorem on the intersection properties of families of convex sets, that allows us to reduce infinite constraints to finitely many, a Calderon-Zygmund decomposition, in order to control the local behaviour of functions and their derivatives, and Ascoli-Arzelà Theorem to ensure compactness.