The structure of Sobolev extension operators

Abstract

Let L-m,L-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 of R-n, denote by L-m,L-p (E) the space of restrictions to E of functions F is an element of L-m,L-p (R-n). It is known that there exist bounded linear maps T : L-m,L-p (E) -> L-m,L-p (R-n) such that T f = f on E for any f is an element of L-m,L-p (E). We show that T cannot have a simple form called “bounded depth”.