1518 CHAPTER 46. SOBOLEV SPACES ON MANIFOLDS

This works because from the definition of γ on C∞(Ω)

and continuity of the map estab-lished above, γ and

(H−1

i

)∗commute and so

γRg ≡l

∑i=1

γ(H−1

i)∗

Rh∗i (ψ ig)

=l

∑i=1

(H−1

i)∗

γRh∗i (ψ ig)

=l

∑i=1

(H−1

i)∗h∗i (ψ ig) = g.

This proves the following theorem about the trace.

Theorem 46.2.2 Let Ω ∈ Cm−1,1. Then there exists a constant, C independent of u ∈W m,p (Ω) and a continuous linear map, γ : W m,p (Ω)→W m− 1

p ,p (∂Ω) such that 46.2.3holds. This map satisfies γu(x) = u(x) for all u ∈C∞

(Ω)

and γ is onto. In the case where

m = 1, there exists a continuous linear map, R : W 1− 1p ,p (∂Ω)→W 1,p (Ω) which has the

property that γRg = g for all g ∈W 1− 1p ,p (∂Ω).

Of course more can be proved but this is all to be presented here.