45.3. INTRINSIC NORMS 1497

Theorem 45.3.7 An equivalent norm for W θ ,p (Rn) for θ ∈ (0,1) is

||u||pW θ ,p(Rn)

=

||u||pLp(Rn)+

n

∑i=1

∫∞

0t(1−θ)p

∣∣∣∣∣∣∣∣Gi (t)u−ut

∣∣∣∣∣∣∣∣pLp

dtt

= ||u||pLp(Rn)+

n

∑i=1

∫∞

0t(1−θ)p

∣∣∣∣∣∣∣∣u(·+ tei)−u(·)t

∣∣∣∣∣∣∣∣pLp

dtt

(45.3.14)

Note it is obvious from 45.3.13 that a Lipschitz map takes W θ ,p (Rn) to W θ ,p (Rn) andis continuous.

The above description in Theorem 45.3.7 also makes possible the following corollary.

Corollary 45.3.8 W θ ,p (Rn) is reflexive.

Proof: Let u ∈W θ ,p (Rn). For each i = 1,2, · · · ,n, define for t > 0,

∆iu(t)(x)≡u(x+ tei)−u(x)

t

Then by Theorem 45.3.7,

∆iu ∈ Lp ((0,∞) ;Lp (Rn) ,µ)≡ Y

whereµ (E)≡

∫E

t(1−θ)pt−1dt.

Clearly the measure space is σ finite and so Y is reflexive by Corollary 21.8.9 on Page687. Also ∆i is a closed operator whose domain is W θ ,p (Rn). To see this, supposeun ∈W θ ,p (Rn) and un → u in Lp (Rn) while ∆iun → g in Y. Then in particular ||∆iun||Yis bounded. Now by Fatou’s lemma,∫

∞

0t(1−θ)p

∣∣∣∣∣∣∣∣u(·+ tei)−u(·)t

∣∣∣∣∣∣∣∣pLp(Rn)

dtt≤

lim infn→∞

∫∞

0t(1−θ)p

∣∣∣∣∣∣∣∣un (·+ tei)−un (·)t

∣∣∣∣∣∣∣∣pLp(Rn)

dtt< ∞.

Letting ∆⃗≡ (∆1,∆2, · · · ,∆n) , it follows from similar reasoning that ∆⃗ is a closed oper-ator mapping W θ ,p (Rn) to Y n. Therefore(

id, ∆⃗)(

W θ ,p (Rn))⊆ Lp (Rn)×Y n

and is a closed subspace of the reflexive space Lp (Rn)×Y n. With the norm in Lp (Rn)×Y n

given as the sum of the norms of the components, it follows the mapping(

id, ∆⃗)

is a