The Structure of Sobolev Extension Operators

TL;DR

This paper investigates the structure of Sobolev extension operators, demonstrating that such operators cannot be represented in a simple, bounded depth form, which has implications for understanding their complexity.

Contribution

The paper proves that Sobolev extension operators cannot be expressed with a simple bounded depth structure, revealing limitations in their potential representations.

Findings

Bounded linear extension operators exist for Sobolev spaces.

Such operators cannot be simplified into bounded depth forms.

Implications for the complexity of Sobolev extension problems.

Abstract

Let $L^{m,p}(\R^n)$ denote the Sobolev space of functions whose $m$-th derivatives lie in $L^p(\R^n)$, and assume that $p>n$. For $E \subset \R^n$, denote by $L^{m,p}(E)$ the space of restrictions to $E$ of functions $F \in L^{m,p}(\R^n)$. It is known that there exist bounded linear maps $T : L^{m,p}(E) \rightarrow L^{m,p}(\R^n)$ 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."