37.1. EMBEDDING THEOREMS FOR W m,p (Rn) 1281

Proof: If |α| ≤ m, then Dα u ∈W j,p (Rn) and so by Corollary 37.1.11, Dα u ∈ Lq (Rn)where q is given above. This means u ∈W m,q (Rn).

The above corollaries imply yet another interesting corollary which involves embed-dings in the Holder spaces.

Corollary 37.1.13 Suppose jp< n< ( j+1) p and let m be a positive integer. Let U be anybounded open set in Rn. Then letting rU denote the restriction to U , rU : W m+ j,p (Rn)→Cm−1,λ

(U)

is continuous for every λ ≤ λ 0 ≡ ( j+1)− np and if λ < ( j+1)− n

p , then rUis compact.

Proof: From Corollary 37.1.12 W m+ j,p (Rn)⊆W m,q (Rn) where q is given by 37.1.16.Therefore,

npn− jp

> n

and so by Corollary 37.1.7, W m,q (Rn)⊆Cm−1,λ(U)

for all λ satisfying

0 < λ < 1− (n− jp)nnp

=p( j+1)−n

p= ( j+1)− n

p.

The assertion about compactness follows from the compactness of the embedding of

Cm−1,λ 0(U)

into Cm−1,λ(U)

for λ < λ 0. See Lemma 37.1.6.There are other embeddings of this sort available. You should see Adams [1] for a

more complete listing of these. Next are some theorems about compact embeddings. Thisrequires some consideration of which subsets of Lp (U) are compact. The main theorem isthe following. See [1].

Theorem 37.1.14 Let K be a bounded subset of Lp (U) and suppose that for all ε > 0,there exist a δ > 0 such that if |h|< δ , then∫

Rn|ũ(x+h)− ũ(x)|p dx < ε

p (37.1.17)

Suppose also that for each ε > 0 there exists an open set, G ⊆U such that G is compactand for all u ∈ K, ∫

U\G|u(x)|p dx < ε

p (37.1.18)

Then K is precompact in Lp (Rn).

Proof: To save fussing first consider the case where U = Rn so that ũ = u. Supposethe two conditions hold and let φ k be a mollifier of the form φ k (x) = knφ (kx) wherespt(φ)⊆ B(0,1) . Consider

Kk ≡ {u∗φ k : u ∈ K} .

and verify the conditions for the Ascoli Arzela theorem for these functions defined on G.Say ||u||p ≤M for all u ∈ K.