1274 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

Theorem 37.1.1 Suppose u,u,i ∈ Lploc (R

n) for i = 1, · · · ,n and p > n. Then u has a repre-sentative, still denoted by u, such that for all x,y ∈Rn,

|u(x)−u(y)| ≤C(∫

B(x,2|y−x|)|∇u|pdz

)1/p

|x−y|(1−n/p). (37.1.13)

This amazing result shows that every u ∈W m,p (Rn) has a representative which is con-tinuous provided p > n.

Using the above inequality, one can give an important embedding theorem.

Definition 37.1.2 Let X ,Y be two Banach spaces and let f : X → Y be a function. Then fis a compact map if whenever S is a bounded set in X , it follows that f (S) is precompactin Y .

Theorem 37.1.3 Let U be a bounded open set and for u a function defined on Rn, letrU u(x)≡ u(x) for x∈U . Then if p> n, rU :W 1,p (Rn)→C

(U)

is continuous and compact.

Proof: First suppose uk → 0 in W 1,p (Rn) . Then if rU uk does not converge to 0, itfollows there exists a sequence, still denoted by k and ε > 0 such that uk→ 0 in W 1,p (Rn)but ||rU uk||∞ ≥ ε. Selecting a further subsequence which is still denoted by k, you can alsoassume uk (x)→ 0 a.e. Pick such an x0 ∈U where this convergence takes place. Then from37.1.13, for all x ∈U ,

|uk (x)| ≤ |uk (x0)|+C ||uk||1,p,Rn diam(U)

showing that uk converges uniformly to 0 on U contrary to ||rU uk||∞ ≥ ε. Therefore, rU iscontinuous as claimed.

Next let S be a bounded subset of W 1,p (Rn) with ||u||1,p < M for all u ∈ S. Then foru ∈ S

rpmn ([|u|> r]∩U)≤∫[|u|>r]∩U

|u|p dmn ≤Mp

and somn ([|u|> r]∩U)≤ Mp

rp .

Now choosing r large enough, Mp/rp < mn (U) and so, for such r, there exists xu ∈U suchthat |u(xu)| ≤ r. Therefore from 37.1.13, whenever x ∈U,

|u(x)| ≤ |u(xu)|+CM diam(U)1−n/p

≤ r+CM diam(U)1−n/p

showing that {rU u : u ∈ S} is uniformly bounded. But also, for x,y ∈U ,37.1.13 implies

|u(x)−u(y)| ≤CM |x−y|1−np

showing that {rU u : u ∈ S} is equicontinuous. By the Ascoli Arzela theorem, it followsrU (S) is precompact and so rU is compact.