Cm,ω extension by bounded-depth linear operators

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