Let Lm,p(ℝn) denote the Sobolev space of functions whose m-th derivatives lie in Lp(ℝn), and assume that p > n. For E ⊆ ℝn, denote by L m,p(E) the space of restrictions to E of functions F ε L m,p(ℝn). It is known that there exist bounded linear maps T : Lm,p(E) → Lm,p(ℝn) such that Tf = f on E for any f ε Lm,p(E). We show that T cannot have a simple form called "bounded depth".
All Science Journal Classification (ASJC) codes
- Linear operators
- Sobolev spaces
- Whitney extension problem