21.10. SOME EMBEDDING THEOREMS 693

Usually the thing of interest in this theorem is the case where V = W = U = R. How-ever, the more general version to be presented is interesting I think. Of course closed andbounded sets are compact in R so the usual case works as a special case of what is about tobe presented.

Theorem 21.10.4 Let V ⊆W ⊆U where these are Banach spaces such that the injectionmap of V into W is compact and the injection map of W into U is continuous. Let Ω be anopen set in Rm and let A be a bounded subset of Lp (Ω;V ) and suppose that for all ε > 0,there exist a δ > 0 such that if |h|< δ , then for ũ denoting the zero extension of u off Ω,∫

Rm∥ũ(x+h)− ũ(x)∥p

U dx < εp (21.10.42)

Suppose also that for each ε > 0 there exists an open set, Gε ⊆ Ω such that Gε ⊆ Ω iscompact and for all u ∈A , ∫

Ω\Gε

∥u(x)∥pW dx < ε

p (21.10.43)

Then A is precompact in Lp (Rn;W ).

Proof: Let ∞ > M ≥ supu∈Lp(Ω;V ) ∥u∥pLp(Ω;V )

. Let {ψn} be a mollifier with support inB(0,1/n). I need to show that A has an η net in Lp (Ω;W ) for every η > 0. Supposefor some η > 0 it fails to have an η net. Without loss of generality, let η < 1. Then by21.10.43, it follows that for small enough ε > 0,Aε ≡

{uXGε

: u ∈A}

fails to have anη/2 net. Indeed, pick ε small enough that for all u ∈A ,∥∥∥uXGε

−u∥∥∥

Lp(Ω;W )<

η

5

Then if{

ukXGε

}r

k=1is an η/2 net for Aε , so that ∪r

k=1B(

ukXGε, η

2

)⊇ Aε , then for

w ∈A , wXGε∈ B

(ukXGε

, η

2

)for some uk. Hence,

∥w−uk∥Lp(Ω;W ) ≤∥∥∥w−wXGε

∥∥∥Lp(Ω;W )

+∥∥∥wXGε

−ukXGε

∥∥∥Lp(Ω;W )

+∥∥∥ukXGε

−uk

∥∥∥Lp(Ω;W )

≤ η

5+

η

2+

η

5< η

and so {uk}rk=1 would be an η net for A which is assumed to not exist.

Pick this ε in all that follows. By compactness, Lemma 21.10.1, there exists Cη suchthat for all u ∈V,

∥u∥pW ≤

η

50(2p−1)M∥u∥p

V +Cη ∥u∥pU (21.10.44)

Let Aεn consist of Aεn ≡{

uXGε∗ψn : u ∈A

}. I want to show that Aεn satisfies the

conditions for Theorem 21.10.3.