The classical Whitney extension theorem states that every function in Lip, , closed, , a non-negative integer, can be extended to a function in Lip. Her Lip stands for the class of functions which on have continuous partial derivatives up to order satisfying certain Lipschitz conditions in the supremum norm. We formulate and prove a similar theorem in the -norm.
The restrictions to , , of the Bessel potential spaces in and the Besov or generalized Lipschitz spaces in 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 is replaced by a closed set 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 . It also gives a contribution to the restriction and extension problem corresponding to the case with equal to the closure of a domain in .
D’après le théorème de prolongement classique de Whitney on peut prolonger toute fonction dans Lip, , fermé, , un nombre entier non-négatif, à une fonction dans Lip. Ici on désigne par Lip l’espace des fonctions sur avec des dérivées partielles continues jusqu’à l’ordre qui satisfont certaines conditions de Lipschitz dans la norme supremum. Nous formons et montrons un théorème analogue dans la norme .
Les restrictions à , , des espaces potentiels besseliens dans 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 est remplacé par un ensemble fermé d’une sorte beaucoup plus générale que les ensembles considérés précédemment. Notre méthode donne une démonstration nouvelle aussi dans le cas . Elle donne aussi une contribution au problème de restriction et prolongement correspondant au cas avec égal à la fermeture d’un domaine dans .
@article{AIF_1978__28_1_139_0, author = {Jonsson, Alf and Wallin, Hans}, title = {A {Whitney} extension theorem in $L^p$ and {Besov} spaces}, journal = {Annales de l'Institut Fourier}, pages = {139--192}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {28}, number = {1}, year = {1978}, doi = {10.5802/aif.684}, zbl = {0369.46031}, mrnumber = {81c:46024}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.684/} }
TY - JOUR AU - Jonsson, Alf AU - Wallin, Hans TI - A Whitney extension theorem in $L^p$ and Besov spaces JO - Annales de l'Institut Fourier PY - 1978 SP - 139 EP - 192 VL - 28 IS - 1 PB - Institut Fourier PP - Grenoble UR - https://aif.centre-mersenne.org/articles/10.5802/aif.684/ DO - 10.5802/aif.684 LA - en ID - AIF_1978__28_1_139_0 ER -
%0 Journal Article %A Jonsson, Alf %A Wallin, Hans %T A Whitney extension theorem in $L^p$ and Besov spaces %J Annales de l'Institut Fourier %D 1978 %P 139-192 %V 28 %N 1 %I Institut Fourier %C Grenoble %U https://aif.centre-mersenne.org/articles/10.5802/aif.684/ %R 10.5802/aif.684 %G en %F AIF_1978__28_1_139_0
Jonsson, Alf; Wallin, Hans. A Whitney extension theorem in $L^p$ and Besov spaces. Annales de l'Institut Fourier, Volume 28 (1978) no. 1, pp. 139-192. doi : 10.5802/aif.684. https://aif.centre-mersenne.org/articles/10.5802/aif.684/
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