Whitney’s extension theorem revisited

March 9, 2010

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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