Whitney’s extension theorem revisited

I (very surprisingly) bumped into Charles Fefferman at Northwestern this afternoon…Hence we talked math for a little bit. Among other things I mentioned that I’ve been trying to extend C^1 functions to the disc volume-preservingly. After trying on the board for a while, he laughs out loud when he saw that this may be obtained applying his favorite Whitney’s extension theorem. (I’ll discuss what he did later in the poster)

Mean while, it’s a pity that I’ve never written a post on Whitney’s extension theorem, hence here it is~

Given a compact subset K in \mathbb{R}^n and a function f: K \rightarrow \mathbb{R}, when can we extend it to a C^r function on the whole \mathbb{R}^n?

First we note that there are obvious cases for which this can’t be done: for example, if we take E to be a segment in \mathbb{R}^2 and f a one-variable function of lower regularity than r, then of course there are no way to find a C^r extension.

Hence it’s only reasonable to restrict our attention to those f that has ‘candidate derivatives’ of all orders no larger than r at all points in E.

i.e. For any k-fold subscript d= (d_1, d_2, \cdots, d_k) with d_1+d_2+ \cdots +d_k \leq r (we will denote d_1+d_2+ \cdots +d_k = |d|, there is a continuous function f_d: K \rightarrow \mathbb{R} with the following property:

For all x_o \in K, \displaystyle f_d(x) = \sum_{|l| \leq r-|d|} \frac{f_{l+d}(x)}{l!}(x-x_0)^l+R_d(x, x_0)  where R_d(x, x_0) \sim o(|x-x_0|^{r-|d|}) as x \rightarrow x_0 and is uniform in x_0.

i.e. The functions f_\alpha are compatible as Taylor coefficients of some C^r function on \mathbb{R}^n, which is absolutely necessary for a C^r extension to exist.

Whitney’s extension theorem: (classical version)

Suppose a set of functions f_\alpha with all multi-index | \alpha | \leq r satisfying the above Taylor condition at all points in $K$. Then there is a C^r function \hat{f}: \mathbb{R}^n \rightarrow \mathbb{R} s.t. \hat{f}|_K = f_{\bar{0}} and for all \alpha \leq r, (D^\alpha \hat{f})|_K = f_\alpha. Furthermore, \hat{f} can be taken real analytic on \mathbb{R}^n \backslash K.

This is indeed the best one could hope for. i.e. there is a C^r extension whenever possible, furthermore the extension is at worst C^r at the points which it is given to be only C^r and much better (analytic) everywhere else.

However, sometimes we would like to control the C^r norm of the resulting function in terms of the C^r norm of the function on K.

Theorem: (Fefferman)

For any n, \ r, there exists C such that the extension ||\hat{f}|| \leq C \cdot ||f|| where the norm is the C^r norm.

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