692 CHAPTER 24. THE BOCHNER INTEGRAL

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 by24.43, it follows that for small enough ε > 0,Aε ≡

{uXGε

: u ∈A}

fails to have an η/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 24.11.1, there exists Cη suchthat for all u ∈V,

∥u∥pW ≤

η

50(2p−1)M∥u∥p

V +Cη ∥u∥pU (24.44)

Let Aεn consist of Aεn ≡{

uXGε∗ψn : u ∈A

}. I want to show that Aεn satisfies the

conditions for Theorem 24.11.3.

Lemma 24.11.5 For each n, Aεn satisfies the conditions of Theorem 24.11.3.

Proof: First consider the equicontinuity condition of that theorem. It suffices to showthat if η > 0 then there exists δ > 0 such that if |h|< δ , then for any u ∈A and x ∈ Gε ,∥∥∥uXGε

∗ψn (x+h)−uXGε∗ψn (x)

∥∥∥U< η

Always assume |h|< dist(Gε ,Ω

C), and x∈Gε . Also assume that |h| is small enough that(∫

Rm

∣∣∣(XGε(x−y+h)−XGε

(x−y))

ψn (y)∣∣∣p′ dz

)1/p′

=

(∫Rm

∣∣∣(XGε(z+h)−XGε

(z))

ψn (x−z)∣∣∣p′ dz

)1/p′

<η

2M(24.45)

This can be obtained because by Holder’s inequality,(∫Rm

∣∣∣(XGε(z+h)−XGε

(z))

ψn (x−z)∣∣∣p′ dz

)1/p′

≤(∫

Rm

∣∣∣XGε(z+h)−XGε

(z)∣∣∣2p′

dz) 1

2p′(∫

Rmψn (x−z)

2p′ dz) 1

2p′

which is small independent of x for |h| small enough, thanks to continuity of translation inL2p′ (Rm). Then