1300 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

=m

∑j=0

C j

∣∣∣∣∣∣ηE j

((h−1

j

)∗(uψ j

))∣∣∣∣∣∣k,p,Rn

≤m

∑j=0

C j

∣∣∣∣∣∣E j

((h−1

j

)∗(uψ j

))∣∣∣∣∣∣k,p,Rn

≤m

∑j=0

C j

∣∣∣∣∣∣(h−1j

)∗(uψ j

)∣∣∣∣∣∣k,p,h j(U∩U j)

≤m

∑j=0

C j

∣∣∣∣∣∣uψ j

∣∣∣∣∣∣k,p,U∩Uk

≤m

∑j=0

C j ||u||k,p,U∩Uk≤

(m

∑j=0

C j

)||u||k,p,U .

Similarly E : W l,p (U)→W l,p (U) for l ≤ k. This proves the theorem.

Definition 37.4.6 When E is a linear continuous map from W l,p (U) to W l,p (Rn) for eachl ≤ k. it is called a strong (k, p) extension map.

There is also a very easy sort of extension theorem for the space, W m,p0 (U) which does

not require any assumptions on the boundary of U other than mn (∂U) = 0. First here isthe definition of W m,p

0 (U) .

Definition 37.4.7 Denote by W m,p0 (U) the closure of C∞

c (U) in W m,p (U) .

Theorem 37.4.8 For u ∈W m,p0 (U) , define

Eu(x)≡{

u(x) if x ∈U0 if x /∈U

Then E is a strong (k, p) extension map.

Proof: Letting l ≤ m, it is clear that for |α| ≤ l,

Dα Eu =

{Dα u for x ∈U0 for x /∈U .

This follows because, since mn (∂U) = 0 it suffices to consider φ ∈C∞c (U) and

φ ∈C∞c

(UC).

Therefore, ||Eu||l,p,Rn = ||u||l,p,U .There are many other extension theorems and if you are interested in pursuing this

further, consult Adams [1]. One of the most famous which is discussed in this reference isdue to Calderon and depends on the theory of singular integrals.