The structure of Sobolev extension operators

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”.

Locations

• Revista Matemática Iberoamericana - View - PDF
• arXiv (Cornell University) - View - PDF