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

Annales de l'institut Fourier (1978)

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

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topJonsson, 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, $k< \beta \le k+1$, $k$ 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 $\{\bf R\}^n$.},

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, $k< \beta \le k+1$, $k$ 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 ${\bf R}^n$.

LA - eng

UR - http://eudml.org/doc/74344

ER -

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