Kenneth Kuttler

1296 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

Proof: Let uk→ 0 in W m+1,p (U) . Then it follows that for each |α| ≤ m, Dα uk→ 0 inW 1,p (U) . Therefore,

E (Dα uk)→ 0 in W 1,p (Rn) .

Then from Morrey’s inequality, 37.1.13 on Page 1274, if λ ≤ 1− np and |α|= m

ρλ (E (Dα uk))≤C ||E (Dα uk)||1,p,Rn diam(U)1− np−λ .

Therefore, ρλ (E (Dα uk)) = ρλ (Dα uk)→ 0. From Theorem 37.3.4 it follows that for |α| ≤

m, ||Dα uk||∞→ 0 and so ||uk||m,λ → 0. This proves the claim about continuity. The claimabout compactness for λ < 1− n

p follows from Lemma 37.1.6 on Page 1275 and this.

(Bounded in W m,p (U)id→ Bounded in Cm,1− n

p(U) id→ Compact in Cm,λ

(U).)

Theorem 37.3.6 Suppose jp < n < ( j+1) p and let m be a positive integer. Let U beany bounded open set in Rn which has a (1, p) extension operator for each p ≥ 1 and thesegment property. Then id ∈L

(W m+ j,p (U) ,Cm−1,λ

(U))

for every λ ≤ λ 0 ≡ ( j+1)− np

and if λ < ( j+1)− np , id is compact.

Proof: From Corollary 37.3.3 W m+ j,p (U) ⊆W m,q (U) where q is given by 37.3.34.Therefore,

npn− jp

> n

and so by Corollary 37.3.5, W m,q (U)⊆Cm−1,λ(U)

for all λ satisfying

0 < λ < 1− (n− jp)nnp

=p( j+1)−n

p= ( j+1)− n

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, Lemma 37.1.6 on Page 1275.

37.4 More Extension TheoremsThe theorem about the existence of a (1, p) extension is all that is needed to obtain gen-eral embedding theorems for Sobolev spaces. However, a more general theory is neededin order to tie the theory of Sobolev spaces presented thus far to a very appealing descrip-tion using Fourier transforms. First the problem of extending W k,p (H) to W k,p (Rn) isconsidered for H− a half space

H− ≡ {y ∈ Rn : yn < 0} . (37.4.35)

I am following Adams [1].

1296 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACESProof: Let u, — 0 in W”"*!? (U). Then it follows that for each |a@| < m, D® uz > 0 inWw! (U). Therefore,E(D%u,) > 0in W!? (R").Then from Morrey’s inequality, 37.1.13 on Page 1274, if A < 1— p and |a| =mPp, (E (D%ug)) < C\|E (D% up) || pn diam(U)'~Therefore, p, (E (D%ux)) = Pp, (D% ux) — 0. From Theorem 37.3.4 it follows that for |a| <m, ||Dug||.. — 0 and so ||W4||n,, —> 0. This proves the claim about continuity. The claimabout compactness for A < 1 — p follows from Lemma 37.1.6 on Page 1275 and this.(Bounded in W”? (U) “4 Bounded in C”!~? (U) “4 Compact in C”™ (U).)Theorem 37.3.6 Suppose jp <n < (j+1)p and let m be a positive integer. Let U beany bounded open set in R" which has a (1, p) extension operator for each p > | and thesegment property. Then id € Z (W™*/? (U) som la (U)) for every A < Ay = (f+ 1) - >and ifA < (j+1)- p id is compact.Proof: From Corollary 37.3.3 W"t/-? (U) C W™4(U) where q is given by 37.3.34.Therefore,npn— jpand so by Corollary 37.3.5, W"7(U) Cc"! (U) for all A satisfying>n(n—jp)n _ p(itl)—nnp Pp0<A<1- _!=(Ui+1)The assertion about compactness follows from the compactness of the embedding ofc7—|Ao (V)into C1 (Z) for A < Ao, Lemma 37.1.6 on Page 1275.37.4 More Extension TheoremsThe theorem about the existence of a (1, p) extension is all that is needed to obtain gen-eral embedding theorems for Sobolev spaces. However, a more general theory is neededin order to tie the theory of Sobolev spaces presented thus far to a very appealing descrip-tion using Fourier transforms. First the problem of extending W"? (H) to W*? (IR") isconsidered for H~ a half spaceH” ={ye€R":y, < O}. (37.4.35)Iam following Adams [1].