# The structure of Sobolev extension operators

### Charles Fefferman

Princeton University, United States### Arie Israel

University of Texas at Austin, USA### Garving K. Luli

University of California at Davis, USA

## Abstract

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”.

## Cite this article

Charles Fefferman, Arie Israel, Garving K. Luli, The structure of Sobolev extension operators. Rev. Mat. Iberoam. 30 (2014), no. 2, pp. 419–429

DOI 10.4171/RMI/787