37.4. MORE EXTENSION THEOREMS 1299
Corollary 37.4.4 Let {k1, · · · ,kr} ⊆ {1, · · · ,n} where the ki are distinct and let
H−k1···kr≡ H−k1
∩H−k2∩·· ·∩H−kr
. (37.4.37)
Then there exists E : W k,p(
H−k1···kr
)→W k,p (Rn) such that E is linear and continuous.
Proof: Follow the above argument with minor modifications to first extend from H−k1···kr
to H−k1···kr−1and then from from H−k1···kr−1
to H−k1···kr−2etc.
This easily implies the ability to extend off bounded open sets which near their bound-aries look locally like an intersection of half spaces.
Theorem 37.4.5 Let U be a bounded open set and suppose U0,U1, · · · ,Um are open setswith the property that U ⊆∪m
k=0Uk,U0⊆U, and ∂U ⊆∪mk=1Uk. Suppose also there exist one
to one and onto functions, hk :Rn→Rn, hk (Uk ∩U) =Wk where Wk equals the intersectionof a bounded open set with a finite intersection of half spaces, H−k1···kr
, as in 37.4.37 suchthat hk (∂U ∩Uk)⊆ ∂H−k1···kr
. Suppose also that for all |α| ≤ k−1,
Dα hk and Dα h−1k
exist and are Lipschitz continuous. Then letting W be an open set which contains U , thereexists E : W k,p (U)→W k,p (W ) such that E is a linear continuous map from W l,p (U) toW l,p (W ) for each l ≤ k.
Proof: Let ψ j ∈C∞c (U j), ψ j (x) ∈ [0,1] for all x ∈Rn, and ∑
mj=0 ψ j (x) = 1 on U . This
is a C∞ partition of unity on U . By Theorem 37.0.14(
h−1j
)∗uψ j ∈W k,p (Wj) . By the
assumption that h j (∂U ∩U j) ⊆ ∂H−k1···kr, the zero extension of
(h−1
j
)∗uψ j to the rest of
H−k1···krresults in an element of W k,p
(H−k1···kr
). Apply Corollary 37.4.4 to conclude there
exists E j : W k,p(
H−k1···kr
)→W k,p (Rn) which is continuous and linear. Abusing notation
slightly, by using(
h−1j
)∗uψ j as the above zero extension, it follows E j
((h−1
j
)∗uψ j
)∈
W k,p (Rn) . Now let η be a function in C∞c (h(W )) such that η (y) = 1 on h
(U). Then
Define
Eu≡m
∑j=0
h∗jηE j
((h−1
j
)∗(uψ j
)).
Clearly Eu(x) = u(x) if x ∈U. It is also clear that E is linear. It only remains to verify Eis continuous. In what follows, C j will denote a constant which is independent of u whichmay change from line to line. By Theorem 37.0.14,
||Eu||k,p,W ≤m
∑j=0
∣∣∣∣∣∣h∗jηE j
((h−1
j
)∗(uψ j
))∣∣∣∣∣∣k,p,W
≤m
∑j=0
C j
∣∣∣∣∣∣ηE j
((h−1
j
)∗(uψ j
))∣∣∣∣∣∣k,p,h(W )