38.6. SOBOLEV SPACES ON MANIFOLDS 1329

Proof: First consider the claim that γ ∈ L(Ht (Ω) ,Ht−1/2 (Γ)

). This involves first

showing that for u ∈ Ht (Ω) ,γu ∈ Ht−1/2 (Γ) . To do this, use the above definition.

h∗j(

ψ j (γu))

=q

∑i=1

h∗j(

ψ jg∗i (γH∗i (uψ i))

)=

q

∑i=1

(h∗jψ j

)(h∗j (g

∗i (γH∗i (uψ i)))

)=

q

∑i=1

(h∗jψ j

)(gi ◦h j)

∗ (γH∗i (uψ i)) (38.6.27)

First note that γH∗i (uψ i) ∈ Ht−1/2 (U ′i ). Now gi ◦h j and its inverse, g j ◦hi are both func-tions in Cm,1

(Rn−1

)and

gi ◦h j : U ′j→U ′i .

Therefore, by Lemma 38.3.3 on Page 1312,

(gi ◦h j)∗ (γH∗i (uψ i)) ∈ Ht−1/2 (U ′j)

and∣∣∣∣(gi ◦h j)

∗ (γH∗i (uψ i))∣∣∣∣

Ht−1/2(

U ′j) ≤ Ci j ||γH∗i (uψ i)||Ht−1/2(U ′i )

. Also it follows that

h∗jψ j ∈Cm,1(

U ′j)

and has compact support in U ′j and so by Corollary 38.3.5 on Page 1314(h∗jψ j

)(gi ◦h j)

∗ (γH∗i (uψ i)) ∈ Ht−1/2 (U ′j)and ∣∣∣∣∣∣(h∗jψ j

)(gi ◦h j)

∗ (γH∗i (uψ i))∣∣∣∣∣∣

Ht−1/2(

U ′j)

≤ Ci j∣∣∣∣(gi ◦h j)

∗ (γH∗i (uψ i))∣∣∣∣

Ht−1/2(

U ′j) (38.6.28)

≤ Ci j ||γH∗i (uψ i)||Ht−1/2(U ′i ). (38.6.29)

This shows γu ∈ Ht−1/2 (Γ) because each h∗j(

ψ j (γu))∈ Ht−1/2

(U ′j). Also from

38.6.29 and 38.6.27

||γu||2Ht−1/2(Γ)≤

q

∑j=1

∣∣∣∣∣∣h∗j (ψ j (γu))∣∣∣∣∣∣2

Ht−1/2(

U ′j)

=q

∑j=1

∣∣∣∣∣∣h∗j (ψ j (γu))∣∣∣∣∣∣2

Ht−1/2(

U ′j) = q

∑j=1

∣∣∣∣∣∣∣∣∣∣ q

∑i=1

(h∗jψ j

)(gi ◦h j)

∗ (γH∗i (uψ i))

∣∣∣∣∣∣∣∣∣∣2

Ht−1/2(

U ′j)

≤ Cq

q

∑j=1

q

∑i=1

∣∣∣∣∣∣(h∗jψ j

)(gi ◦h j)

∗ (γH∗i (uψ i))∣∣∣∣∣∣2

Ht−1/2(

U ′j)

≤Cq

q

∑j=1

q

∑i=1

Ci j ||(γH∗i (uψ i))||2Ht−1/2(U ′i )

≤Cq

q

∑i=1||(γH∗i (uψ i))||

2Ht−1/2(U ′i )