1508 CHAPTER 45. TRACES OF SOBOLEV SPACES

where this E pertains to extending W 1,p(U).

C1 || f ||WU≥ ||E f ||W ≥ ||Eu|| ˜W θ ,p(Rn)

≥ ||u|| ˜W θ ,p(U)

Since this is true for every f ∈WU , it follows u ∈ ˜W θ ,p (U) and

||u|| ˜W θ ,p(U)≤C1 ||u||W θ ,p(U) .

This proves the theorem.

Corollary 45.3.14 Let U be a bounded open set with Lipschitz boundary. Then W θ ,p (U)is reflexive.

Proof: From Proposition 45.3.12 and Theorem 45.3.13, there exists an extension oper-ator E : W θ ,p (U)→W θ ,p (Rn) which is continuous. This operator is one to one and contin-uous. Furthermore, ||Eu||W θ ,p(Rn) ≥ ||u||W θ ,p(U) and so E

(W θ ,p (U)

)is closed. Therefore,

by Corollary 21.2.8 on Page 656 and the fact W θ ,p (Rn) is reflexive which was shown inCorollary 45.3.8, it follows W θ ,p (U) is reflexive. This proves the corollary.

There may be other sets U for which the intrinsic norm is an equivalent norm forW θ ,p (U) but this much will suffice. It should be routine to verify that this works for Ua half space for example and the extension argument should be much easier than that pre-sented above. More generally, the assumption that U was bounded in the above extensionargument of Proposition 45.3.12 was never needed except for giving finitely many of thosespecial sets covering the boundary. If you just assumed this at the outset instead of anassumption the set is bounded, the same sort of extension would work.

45.4 Fractional Order Sobolev SpacesNow it is time to define fractional order Sobolev spaces between W m,p and W m+1,p.

Definition 45.4.1 Let m be a nonnegative integer and let s = m + σ where σ ∈ (0,1) .Then W s,p (Ω) will consist of those elements of W m,p (Ω) for which Dα u ∈W σ ,p (Ω) for all|α|= m. The norm is given by the following.

||u||s,p,Ω ≡

(||u||pm,p,Ω + ∑

|α|=m||Dα u||p

σ ,p,Ω

)1/p

.

Corollary 45.4.2 The space, W s,p (Ω) is a reflexive Banach space whenever p > 1.

Proof: From the theory of interpolation spaces, W σ ,p (Ω) is reflexive. This is becauseit is an iterpolation space for the two reflexive spaces, Lp (Ω) and W 1,p (Ω) . (Alternatively,you could use Corollary 45.3.14 in the case where Ω is a bounded open set with Lips-chitz boundary or you could use Corollary 45.3.8 in case Ω = Rn. In addition, the sameideas would work if Ω were any space for which there was a continuous extension mapfrom W σ ,p (Ω) to W σ ,p (Rn) .) Now the formula for the norm of an element in W s,p (Ω)

shows this space is isometric to a closed subspace of W m,p (Ω)×W σ ,p (Ω)k for suitable k.Therefore, from Corollary 21.2.8 on Page 656, W s,p (Ω) is also reflexive.