1272 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

and the sum is always finite. Similarly,

Dα∞

∑k=1

(ψku) =∞

∑k=1

Dα (ψku)

and the sum is always finite. Therefore,

||Dα J−Dα u||Lp(U) =

∣∣∣∣∣∣∣∣∣∣ ∞

∑k=1

Dα wk−Dα (ψku)

∣∣∣∣∣∣∣∣∣∣Lp(U)

≤∞

∑k=1||Dα wk−Dα (ψku)||Lp(U) ≤

∞

∑k=1

ε

2k = ε.

By choosing tk small enough, such an inequality can be obtained for∣∣∣∣∣∣Dβ J−Dβ u∣∣∣∣∣∣

Lp(U)

for each multi index, β such that |β | ≤ m. Therefore, there exists

J ∈C∞c (Rn)

such that||J−u||W m,p(U) ≤ εK

where K equals the number of multi indices no larger than m. Since ε is arbitrary, thisproves the theorem.

Corollary 37.0.12 Let U be an open set which has the segment property. Then W m,p (U) =Xm,p (U) .

Proof: Start with an open covering of U whose sets satisfy the segment condition andobtain a locally finite refinement consisting of bounded sets which are of the sort in theabove theorem.

Now consider a situation where h : U → V where U and V are two open sets in Rn

and Dα h exists and is continuous and bounded if |α| < m− 1 and Dα h is Lipschitz if|α|= m−1.

Definition 37.0.13 Whenever h :U→V, define h∗ mapping the functions which are definedon V to the functions which are defined on U as follows.

h∗ f (x)≡ f (h(x)) .

h : U → V is bilipschitz if h is one to one, onto and Lipschitz and h−1 is also one to one,onto and Lipschitz.

Theorem 37.0.14 Let h : U→V be one to one and onto where U and V are two open sets.Also suppose that Dα h and Dα

(h−1)

exist and are Lipschitz continuous if |α| ≤ m−1 form a positive integer. Then

h∗ : W m,p (V )→W m,p (U)

is continuous, linear, one to one, and has an inverse with the same properties, the inversebeing

(h−1)∗.