1294 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

37.3 General Embedding TheoremsWith the extension theorem it is possible to give a useful theory of embeddings.

Theorem 37.3.1 Let 1 ≤ p < n and 1q = 1

p −1n and let U be any open set for which there

exists a (1, p) extension operator. Then if u∈W 1,p (U) , there exists a constant independentof u such that

||u||Lq(U) ≤C ||u||1,p,U .

If U is bounded and r < q, then id : W 1,p (U)→ Lr (U) is also compact.

Proof: Let E be the (1, p) extension operator. Then by Theorem 37.1.10 on Page 1279

||u||Lq(U) ≤ ||Eu||Lq(Rn) ≤1n√

n(n−1) p(n− p)

||Eu||1,p,Rn

≤ C ||u||1,p,U .

It remains to prove the assertion about compactness. If S⊆W 1,p (U) is bounded then

supu∈S

{||u||1,1,U + ||u||Lq(U)

}< ∞

and so by Theorem 37.1.17 on Page 1286, it follows S is precompact in Lr (U) .This provesthe theorem.

Corollary 37.3.2 Suppose mp < n and U is an open set satisfying the segment conditionwhich has a (1, p) extension operator for all p. Then id ∈ L (W m,p (U) ,Lq (U)) whereq = np

n−mp .

Proof: This is true if m = 1 according to Theorem 37.3.1. Suppose it is true for m−1where m > 1. If u ∈W m,p (U) and |α| ≤ 1, then Dα u ∈W m−1,p (U) so by induction, for allsuch α,

Dα u ∈ Lnp

n−(m−1)p (U) .

Thus, since U has the segment condition, u ∈W 1,q1 (U) where

q1 =np

n− (m−1) p

By Theorem 37.3.1, it follows u ∈ Lq (Rn) where

1q=

n− (m−1) pnp

− 1n=

n−mpnp

.

This proves the corollary.There is another similar corollary of the same sort which is interesting and useful.