37.1. EMBEDDING THEOREMS FOR W m,p (Rn) 1283

net for Kk in C(G). That is, there exist {ui}m

i=1 ⊆ K such that for any u ∈ K,

∣∣∣∣u∗φ k−u j ∗φ k

∣∣∣∣∞<

(ε

m(G+B(0,1)

))1/p

for some j. Letting u ∈ K be given, let u j ∈ {ui}mi=1 ⊆ K be such that the above inequality

holds. Then∣∣∣∣u−u j∣∣∣∣

p ≤ ||u−u∗φ k||p +∣∣∣∣u∗φ k−u j ∗φ k

∣∣∣∣p +∣∣∣∣u j ∗φ k−u j

∣∣∣∣p

< 2ε +∣∣∣∣u∗φ k−u j ∗φ k

∣∣∣∣p

≤ 2ε +

(∫G+B(0,1)

∣∣u∗φ k−u j ∗φ k

∣∣p dx)1/p

+

(∫Rn\(G+B(0,1))

∣∣u∗φ k−u j ∗φ k

∣∣p dx

)1/p

≤ 2ε + ε1/p

+

(∫Rn\(G+B(0,1))

(∫ ∣∣u(x−y)−u j (x−y)∣∣φ k (y)dy

)p

dx

)1/p

≤ 2ε + ε1/p

+∫

φ k (y)

(∫Rn\(G+B(0,1))

(|u(x−y)|+

∣∣u j (x−y)∣∣)p dx

)1/p

dy

≤ 2ε + ε1/p +

∫φ k (y)

(∫Rn\G

(|u(x)|+

∣∣u j (x)∣∣)p dx

)1/p

dy

≤ 2ε + ε1/p +2p−1

∫φ k (y)

(∫Rn\G

(|u(x)|p +

∣∣u j (x)∣∣p)dx

)1/p

dy

≤ 2ε + ε1/p +2p−121/p

ε

and since ε > 0 is arbitrary, this shows that K is totally bounded and is therefore precom-pact.

Now for an arbitrary open set, U and K given in the hypotheses of the theorem, letK̃ ≡ {ũ : u ∈ K} and observe that K̃ is precompact in Lp (Rn) . But this is the same assaying that K is precompact in Lp (U) . This proves the theorem.

Actually the converse of the above theorem is also true [1] but this will not be neededso I have left it as an exercise for anyone interested.

Lemma 37.1.15 Let u ∈W 1,1 (U) for U an open set and let φ ∈C∞c (U) . Then there exists

a constant,C(

φ , ||u||1,1,U),