Revista Matemática Iberoamericana


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Volume 21, Issue 1, 2005, pp. 313–348
DOI: 10.4171/RMI/424

Interpolation and extrapolation of smooth functions by linear operators

Charles Fefferman[1]

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

Let $C^{m , 1} ( \mathbb{R}^n)$ be the space of functions on $\mathbb{R}^n$ whose $m^{\sf th}$ derivatives are Lipschitz 1. For $E \subset \mathbb{R}^n$, let $C^{m , 1} (E)$ be the space of all restrictions to $E$ of functions in $C^{m,1} ( \mathbb{R}^n)$. We show that there exists a bounded linear operator $T: C^{m , 1} (E) \rightarrow C^{m , 1} ( \mathbb{R}^n)$ such that, for any $f \in C^{m , 1} ( E )$, we have $T f = f$ on $E$.

Keywords: Whitney extension problem, linear operators

Fefferman C. Interpolation and extrapolation of smooth functions by linear operators. Rev. Mat. Iberoamericana 21 (2005), 313-348. doi: 10.4171/RMI/424