A Whitney extension theorem in L p and Besov spaces

Alf Jonsson; Hans Wallin

Annales de l'institut Fourier (1978)

  • Volume: 28, Issue: 1, page 139-192
  • ISSN: 0373-0956

Abstract

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The classical Whitney extension theorem states that every function in Lip ( β , F ) , F ⊂ R n , F closed, k < β ≤ k + 1 , k a non-negative integer, can be extended to a function in Lip ( β , R n ) . Her Lip ( β , 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 R d , d < n , of the Bessel potential spaces in R n and the Besov or generalized Lipschitz spaces in 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 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 = 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 R n .

How to cite

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Jonsson, Alf, and Wallin, Hans. "A Whitney extension theorem in $L^p$ and Besov spaces." Annales de l'institut Fourier 28.1 (1978): 139-192. <http://eudml.org/doc/74344>.

@article{Jonsson1978,
abstract = {The classical Whitney extension theorem states that every function in Lip$(\beta ,F)$, $F\subset \{\bf R\}^n$, $F$ closed, $k&lt; \beta \le k+1$, $k$ 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 $\{\bf R\}^d$, $d&lt; n$, 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 $\{\bf R\}^n$.},
author = {Jonsson, Alf, Wallin, Hans},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {1},
pages = {139-192},
publisher = {Association des Annales de l'Institut Fourier},
title = {A Whitney extension theorem in $L^p$ and Besov spaces},
url = {http://eudml.org/doc/74344},
volume = {28},
year = {1978},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 28
IS - 1
SP - 139
EP - 192
AB - The classical Whitney extension theorem states that every function in Lip$(\beta ,F)$, $F\subset {\bf R}^n$, $F$ closed, $k&lt; \beta \le k+1$, $k$ 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 ${\bf R}^d$, $d&lt; n$, 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 ${\bf R}^n$.
LA - eng
UR - http://eudml.org/doc/74344
ER -

References

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