45.4. FRACTIONAL ORDER SOBOLEV SPACES 1509

Theorem 45.4.3 The trace map, γ : W m,p(Rn+

)→W m− 1

p ,p(Rn−1

)is continuous.

Proof: Let f ∈ S, the Schwartz class. Let σ = 1− 1p so that m−

(1p

)= m− 1+σ .

Then from the definition and using f ∈S,

||γ f ||m− 1p ,p,Rn−1 =

(||γ f ||pm−1,p,Rn−1 + ∑

|α|=m−1||Dα

γ f ||p1− 1

p ,p,Rn−1

)1/p

=

(||γ f ||pm−1,p,Rn−1 + ∑

|α|=m−1||γDα f ||p

1− 1p ,p,Rn−1

)1/p

and from Lemma 45.1.4, and the fact that the trace is continuous as a map from W m,p(Rn+

)to W m−1,p

(Rn−1

),

||γ f ||m− 1p ,p,Rn−1 ≤

(C1 || f ||pm,p,Rn

++C2 ∑

|α|=m−1||Dα f ||1,p,Rn

)1/p

≤ C || f ||m,p,Rn+p .

Then using density of S this implies the desired result.With the definition of W s,p (Ω) for s not an integer, here is a generalization of an earlier

theorem.

Theorem 45.4.4 Let h : U → V where U and V are two open sets and suppose h isbilipschitz and that Dα h and Dα h−1 exist and are Lipschitz continuous if |α| ≤ m wherem = 0,1, · · · .and s = m+σ where σ ∈ (0,1) . Then

h∗ : W s,p (V )→W s,p (U)

is continuous, linear, one to one, and has an inverse with the same properties, the inversebeing

(h−1)∗.

Proof: In case m = 0, the conclusion of the theorem is immediate from the generaltheory of trace spaces. Therefore, assume m≥ 1. It follows from the definition that

||h∗u||m+σ ,p,U ≡

[||h∗u||pm,p,U + ∑

|α|=m||Dα (h∗u)||p

σ ,p,U

]1/p

Consider the case when m = 1. Then it is routine to verify that

D jh∗u(x) = u,k (h(x))hk, j (x) .

Let Lk : W 1,p (V )→W 1,p (U) be defined by

Lkv = h∗ (v)hk, j.