The structure of linear extension operators for C^m

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