Whitney extension theorem holds in sharp form

You need to know: Polynomials and functions on n variables, higher order partial derivatives, Lipschitz function.

Background: For a function $f:{\mathbb R}^n\to{\mathbb R}$ and vector $\beta=(\beta_1, \beta_2 \dots, \beta_n)$ of non-negative integers, denote $\partial^\beta f$ the partial derivative $\frac{\partial^{|\beta|}f}{\partial x_1^{\beta_1} \partial x_2^{\beta_2} \dots \partial x_n^{\beta_n}}$ of order $|\beta|=\beta_1 + \beta_2 + \dots + \beta_n$ . Let $C^m({\mathbb R}^n)$ denote the space of functions $f:{\mathbb R}^n\to{\mathbb R}$ whose derivatives of order $\leq m$ are continuous and bounded on ${\mathbb R}^n$ . Also, let $C^{m-1,1}({\mathbb R}^n)$ denote the space of functions whose derivatives of order $m-1$ are Lipschitz with constant 1.

The Theorem: On 14th May 2003, Charles Fefferman submitted to the Annals of Mathematics a paper in which he proved that for any integers $m\geq 1$ and $n\geq 1$ there exists an integer k, depending only on m and n, for which the following holds. Let $f:E\to{\mathbb R}$ be a function defined on an arbitrary subset $E$ of ${\mathbb R}^n$ . Suppose that, for any k distinct points $x_1,\dots, x_k \in E$ there exist polynomials $P_1,\dots, P_k$ on ${\mathbb R}^n$ of degree $m-1$ satisfying (a) $P_i(x_i)=f(x_i)$ for $i=1,\dots, k$ ; (b) $|\partial^\beta P_i(x_i)| \leq\ M$ for $i=1,\dots, k$ and $|\beta|\leq m-1$ ; and (c) $|\partial^\beta (P_i-P_j)(x_i)| \leq M|x_i-x_j|^{m-|\beta|}$ for $i,j=1,\dots, k$ and $|\beta|\leq m-1$ , where M is a constant independent of $x_1,\dots, x_k$ . Then $f$ extends to $C^{m-1,1}$ function on ${\mathbb R}^n$ .

Short context: Given a function $f:E\to{\mathbb R}$ , where E is a subset of ${\mathbb R}^n$ , how can we decide whether f extends to a $C^m({\mathbb R}^n)$ function F on ${\mathbb R}^n$ ? This is a classical question which has been answered by Whitney in 1934, and the result is known as Whitney extension theorem. The Theorem solves a version of this problem in which F is required to be $C^{m-1,1}({\mathbb R}^n)$ .

Links: The original paper is available here. See also Section 5.2 of this book for an accessible description of the Theorem.

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Whitney extension theorem holds in sharp form

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