694 CHAPTER 24. THE BOCHNER INTEGRAL
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 24.11.3 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 thesecond sort are totally bounded while those functions of the first sort don’t. Assume alwaysthat 1/n < dist
(Gε ,Ω
C). Using Minkowski’s inequality,∥∥∥uXGε
−uXGε∗ψn
∥∥∥Lp(Gε ;W)
=
(∫Rm
∥∥∥∥∫Rm
(uXGε
(x)−uXGε(x−y)
)ψn (y)dy
∥∥∥∥p
Wdx)1/p
≤∫
B(0,1/n)ψn (y)
(∫Rm
∥∥∥(uXGε(x)−uXGε
(x−y))∥∥∥p
Wdx)1/p
dy
≤∫
B(0,1/n)ψn (y)
(∫Rm∥(ũ(x)− ũ(x−y))∥p
W dx)1/p
dy
≤∫
B(0,1/n)ψn (y)
( ∫Rm
η
50(2p−1)M∥ũ(x)− ũ(x−y)∥p
V
+Cη ∥ũ(x)− ũ(x−y)∥pU dx
)1/p
dy
≤∫
B(0, 1n )
ψn (y)
( ∫Rm
η
50(2p−1)M2p−12
(∥ũ(x)∥p
V
)dx
+Cη
∫Rm ∥ũ(x)− ũ(x−y)∥p
U dx
)1/p
dy
≤∫
B(0, 1n )
ψn (y)
( ∫Rm
η
25M
(∥ũ(x)∥p
V
)dx
+∫Rm Cη ∥ũ(x)− ũ(x−y)∥p
U dx
)1/p
dy
≤∫
B(0, 1n )
ψn (y)
(η
25+∫Rm
Cη ∥ũ(x)− ũ(x−y)∥pU dx
)1/p
dy
By assumption 24.42, there exists N such that if n≥ N, then |y|< 1n and for all u ∈A ,∥∥∥uXGε
−uXGε∗ψn
∥∥∥Lp(Gε ;W)
≤∫
B(0, 1n )
ψn (y)
(η
25+
η p
8p
)1/p
dy
≤∫
B(0, 1n )
ψn (y)(
η
25+
η
8
)dy =
η
25+
η
8
Recall η < 1.