A Whitney extension theorem in $L^p$ and Besov spaces

Jonsson, Alf ; Wallin, Hans

Annales de l'Institut Fourier, Tome 28 (1978), p. 139-192 / Harvested from Numdam

Résumé

D’après le théorème de prolongement classique de Whitney on peut prolonger toute fonction dans Lip$(\beta ,F)$, $F\subset {\bf R}^n$, $F$ fermé, $k< \beta \le k+1$, $k$ un nombre entier non-négatif, à une fonction dans Lip$(\beta ,{\bf R}^n)$. Ici on désigne par Lip$(\beta ,F)$ l’espace des fonctions sur $F$ avec des dérivées partielles continues jusqu’à l’ordre $k$ qui satisfont certaines conditions de Lipschitz dans la norme supremum. Nous formons et montrons un théorème analogue dans la norme $L^p$.

Les restrictions à ${\bf R}^d$, $d< n$, des espaces potentiels besseliens dans ${\bf R}^n$ et les espaces de Besov ou les espaces de Lipschitz généralisés sont caractérisées par les travaux de plusieurs auteurs (O.V. Besov, E.M. Stein, et d’autres). Nous traitons, pour les espaces de Besov, le cas quand ${\bf R}^d$ est remplacé par un ensemble $F$ fermé d’une sorte beaucoup plus générale que les ensembles considérés précédemment. Notre méthode donne une démonstration nouvelle aussi dans le cas $F={\bf R}^d$. Elle donne aussi une contribution au problème de restriction et prolongement correspondant au cas $d=n$ avec $F$ égal à la fermeture d’un domaine dans ${\bf R}^n$.

Abstract

The classical Whitney extension theorem states that every function in Lip$(\beta ,F)$, $F\subset {\bf R}^n$, $F$ closed, $k< \beta \le k+1$, $k$ a non-negative integer, can be extended to a function in Lip$(\beta ,{\bf R}^n)$. Her Lip$(\beta ,F)$ stands for the class of functions which on $F$ have continuous partial derivatives up to order $k$ satisfying certain Lipschitz conditions in the supremum norm. We formulate and prove a similar theorem in the $L^p$-norm.

The restrictions to ${\bf R}^d$, $d< n$, of the Bessel potential spaces in ${\bf R}^n$ and the Besov or generalized Lipschitz spaces in ${\bf R}^n$ have been characterized by the work of many authors (O.V. Besov, E.M. Stein, and others). We treat, for Besov spaces, the case when ${\bf R}^d$ is replaced by a closed set $F$ of a much more general kind than the sets which have been considered before. Our method of proof gives a new proof also in the case when $F = {\bf R}^d$. It also gives a contribution to the restriction and extension problem corresponding to the case $d=n$ with $F$ equal to the closure of a domain in ${\bf R}^n$.

Détail

Publié le : 1978-01-01

MR 81c:46024 | Zbl 0369.46031

DOI : https://doi.org/10.5802/aif.684

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