# Interpolation and extrapolation of smooth functions by linear operators

### Charles Fefferman

Princeton University, United States

## Abstract

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 $E$.

## Cite this article

Charles Fefferman, Interpolation and extrapolation of smooth functions by linear operators. Rev. Mat. Iberoam. 21 (2005), no. 1, pp. 313–348

DOI 10.4171/RMI/424