C(m, omega) extension by bounded-depth linear operators

Let Cm,wRn be the space of functions on Rn whose m-th derivatives have modulus of continuity o, and Cm,wE the space of restrictions to E ⊆ Rn of functions in Cm,wRn . We show that for any closed set E ⊆ Rn , there exists a linear extension operator T:Cm,wE→ Cm,wRn that assumes a particularly simple form, namely, T has "depth" d depending only on m and n.