1326 CHAPTER 38. SOBOLEV SPACES BASED ON L2
Lemma 38.6.6 Suppose Γ is a Cm,1 manifold and (Γ,Wi,Ui,Γi,hi,gi) are as described inDefinition 38.6.1. Then for t ∈ [m,m+ s), it follows that if u ∈Ht (Γ) , then u ∈Ht (Γk) andthe restriction map is continuous. Also an equivalent norm for Ht (Γk) is given by
|||u|||t ≡ ||h∗ku||Ht (Uk)
.
Proof: Let u ∈ Ht (Γ) and let (Γk,W ′i ,U′i ,Γ′i,h′i,g′i) be the sets and functions which
define what is meant by Γk being a Cm,1 manifold as described in Definition 38.6.1. Alsolet (Γ,Wi,Ui,Γi,hi,gi) be pertain to Γ in the same way and let
{φ j
}be a C∞ partition of
unity for the{
Wj}
. Since the {W ′i } are locally finite, only finitely many can intersect Γk,say {W ′1, · · · ,W ′s} . Also only finitely many of the Wi can intersect Γk, say
{W1, · · · ,Wq
}.
Then letting {ψ ′i} be a C∞ partition of unity subordinate to the {W ′i } .
∞
∑i=1
∣∣∣∣h′∗i (uψ′i)∣∣∣∣
Ht(U ′i )=
s
∑i=1
∣∣∣∣∣∣∣∣∣∣h′∗i
(q
∑j=1
φ juψ′i
)∣∣∣∣∣∣∣∣∣∣Ht(U ′i )
≤s
∑i=1
q
∑j=1
∣∣∣∣∣∣h′∗i φ juψ′i
∣∣∣∣∣∣Ht(U ′i )
=q
∑j=1
s
∑i=1
∣∣∣∣∣∣h′∗i φ juψ′i
∣∣∣∣∣∣Ht(U ′i )
=q
∑j=1
s
∑i=1
∣∣∣∣∣∣(g j ◦h′i)∗h∗jφ juψ
′i
∣∣∣∣∣∣Ht(U ′i )
.
By Lemma 38.3.3 on page 1312, there exists a single constant, C such that the above isdominated by C ∑
qj=1 ∑
si=1
∣∣∣∣∣∣h∗jφ juψ ′i
∣∣∣∣∣∣Ht(U j)
. Now by Corollary 38.3.5 on Page 1314, this
is no larger than
Cq
∑j=1
s
∑i=1
Cψ ′i
∣∣∣∣∣∣h∗jφ ju∣∣∣∣∣∣
Ht(U j)≤C
q
∑j=1
s
∑i=1
∣∣∣∣∣∣h∗jφ ju∣∣∣∣∣∣
Ht(U j)≤C
q
∑j=1
∣∣∣∣∣∣h∗jφ ju∣∣∣∣∣∣
Ht(U j)< ∞.
This shows that u restricted to Γk is in Ht (Γk). It also shows that the restriction map ofHt (Γ) to Ht (Γk) is continuous.
Now consider the norm |||·|||t . For u ∈ Ht (Γk) , let (Γk,W ′i ,U′i ,Γ′i,h′i,g′i) be sets and
functions which define an atlas for Γk. Since the {W ′i } are locally finite, only finitely manycan have nonempty intersection with Γk, say {W1, · · · ,Ws} . Thus i ≤ s for some finite s.The problem is to compare |||·|||t with ||·||Ht (Γk)
. As above, let {ψ ′i} denote a C∞ partition
of unity subordinate to the{
W ′j}
. Then
|||u|||t ≡ ||h∗ku||Ht (Uk)
=
∣∣∣∣∣∣∣∣∣∣h∗k s
∑j=1
ψ′ju
∣∣∣∣∣∣∣∣∣∣Ht (Uk)
≤s
∑j=1
∣∣∣∣∣∣h∗k (ψ′ju)∣∣∣∣∣∣
Ht (Uk)
=s
∑j=1
∣∣∣∣∣∣(g′j ◦hk)∗h′∗j
(ψ′ju)∣∣∣∣∣∣
Ht (Uk)≤C
s
∑j=1
∣∣∣∣∣∣h′∗j (ψ′ju)∣∣∣∣∣∣
Ht(
U ′j) .