21.10. SOME EMBEDDING THEOREMS 695

You make the obvious change here in case p = 1. Instead of the above, you would have

≤∫Rm∥ũ(z+h)− ũ(z)∥U dz2∥ψn∥∞

Since Lebesgue measure is translation independent, there is a constant Cn such that theabove is

≤Cn

(∫Rm∥ũ(z+h)− ũ(z)∥p

U

)1/p

< η/2

and this holds for all u ∈ A . As for the second integral in 21.10.46, from 21.10.45, itfollows that this term is no larger than

≤(∫

Rm∥ũ(z)∥p

U dz)1/p(∫

Rm

(∣∣∣XGε(z+h)−XGε

(z)∣∣∣ψn (x− z)

)p′

dz)1/p′

and by 21.10.45,

< Mη

2M=

η

2

Thus, if δ < dist(Gε ,Ω

C)

and 21.10.45 holds, then for all u ∈A , when |h|< δ ,∥∥∥uXGε∗ψn (x+h)−uXGε

∗ψn (x)∥∥∥

U< η

and so the desired equicontinuity condition holds for Aεn. Note that δ does depend on nbut for each n, things work out well.

I also need to verify that the functions in Aεn are uniformly bounded. For x ∈ Gε andu ∈A , ∥∥∥uXGε

∗ψn (x)∥∥∥

V≤∫

Gε

∥u(z)∥ψn (x− z)dz

≤(∫

Ω

∥u(z)∥p dz)1/p(∫

Ω

ψn (x− z)p′)1/p′

≤MCn

Now is a general statement about norms, indicating that the Lp norm is no more than aconstant times the norm involving the maximum.(∫

Gε

∥v(x)∥pW dx

)1/p

≤ maxx∈Gε

∥v(x)∥W m(Gε

)≡ m

(Gε

)∥v∥W,∞

It follows from Theorem 21.10.3 that for every η > 0, there exists a η net in C(Gε ;W

)for

Aεn, this for each n. Then from the above inequality, it follows that for each η , there existsan η net in Lp

(Gε ;W

)for Aεn.

Recall also, from the assumption that the theorem is not true, Aε ≡{

uXGε: u ∈A

}has no η/2 net in Lp

(Gε ;W

). Next I estimate the distance in Lp

(Gε ;W

)between uXGε

for u ∈A and uXGε∗ψn. The idea is that for each n,Aεn has an η/8 net and for n large

enough, uXGεis close to uXGε

∗ψn so a contradiction will result if the functions of the