Revista Matemática Iberoamericana

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Volume 30, Issue 2, 2014, pp. 419–429
DOI: 10.4171/RMI/787

Published online: 2014-07-08

The structure of Sobolev extension operators

Charles Fefferman[1], Arie Israel[2] and Garving K. Luli[3]

(1) Princeton University, United States
(2) University of Texas at Austin, USA
(3) University of California at Davis, USA

Let $L^{m,p}(\mathbb{R}^n)$ denote the Sobolev space of functions whose $m$-th derivatives lie in $L^p(\mathbb{R}^n)$, and assume that $p>n$. For $E \subseteq \mathbb{R}^n$, denote by $L^{m,p}(E)$ the space of restrictions to $E$ of functions $F \in L^{m,p}(\mathbb{R}^n)$. It is known that there exist bounded linear maps $T \colon L^{m,p}(E) \rightarrow L^{m,p}(\mathbb{R}^n)$ such that $Tf = f$ on $E$ for any $f \in L^{m,p}(E)$. We show that $T$ cannot have a simple form called “bounded depth”.

Keywords: Whitney extension problem, linear operators, Sobolev spaces

Fefferman Charles, Israel Arie, Luli Garving: The structure of Sobolev extension operators. Rev. Mat. Iberoam. 30 (2014), 419-429. doi: 10.4171/RMI/787