Sobolev Extension By Linear Operators

Abstract

Let $L^{m,p}({\mathbb{R}}^n)$ be the Sobolev space of functions with $m^{th}$ derivatives lying in $L^p({\mathbb{R}}^n)$. Assume that $n<p<\infty$. For $E\subset {\mathbb{R}}^n$, let $L^{m,p}(E)$ denote the space of restrictions to $E$ of functions in $L^{m,p}({\mathbb{R}}^n)$. We show that there exists a bounded linear map $T:L^{m,p}(E)\to L^{m,p}({\mathbb{R}}^n)$ such that, for any $f\in L^{m,p}(E)$, we have $Tf=f$ on $E$. We also give a formula for the order of magnitude of $\Vert f{\Vert }_{L^{m,p}(E)}$ for a given $f:E\to \mathbb{R}$ when $E$ is finite.

98 pages

References in corpus (1)

• On The Sum Of A Sobolev Space And A Weighted $L_P$-Space

Cited by in corpus (2)

• Sobolev $L_p^2$-functions on closed subsets of $R^2$
• On The Sum Of A Sobolev Space And A Weighted $L_P$-Space