Revista Matemática Iberoamericana

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Volume 21, Issue 2, 2005, pp. 577–688
DOI: 10.4171/RMI/430

A Generalized Sharp Whitney Theorem for Jets

Charles Fefferman[1]

(1) Department of Mathematics, Princeton University, Fine Hall, Washington Road, NJ 08544-1000, Princeton, USA

Suppose that, for each point $x$ in a given subset $E \subset \mathbb{R}^n$, we are given an $m$-jet $f(x)$ and a convex, symmetric set $\sigma(x)$ of $m$-jets at $x$. We ask whether there exist a function $F \in C^{m , \omega} ( \mathbb{R}^n )$ and a finite constant $M$, such that the $m$-jet of $F$ at $x$ belongs to $f ( x ) + M \sigma ( x )$ for all $x \in E$. We give a necessary and sufficient condition for the existence of such $F , M$, provided each $\sigma(x)$ satisfies a condition that we call "Whitney $\omega$-convexity''.

Keywords: Extension problems, Whitney convexity, Whitney $\omega$-convexity

Fefferman Charles: A Generalized Sharp Whitney Theorem for Jets. Rev. Mat. Iberoamericana 21 (2005), 577-688. doi: 10.4171/RMI/430