The Structure of Linear Extension Operators for $C^m$

For any subset E ⊂ Rn, let Cm(E) denote the Banach space of restrictions to E of functions F ∈ Cm(Rn). It is known that there exist bounded linear maps T : Cm(E) −→ Cm(Rn) such that Tf = f on E for any f ∈ Cm(E). We show that T can be taken to have a simple form, but cannot be taken to have an even simpler form.