1330 CHAPTER 38. SOBOLEV SPACES BASED ON L2

≤Cq

q

∑i=1||H∗i (uψ i)||

2Ht (Ri(Wi∩Ω)) ≤Cq

q

∑i=1||uψ i||

2Ht (Wi∩Ω) ≤Cq ||u||2Ht (Ω) .

Does γ satisfy 38.6.26? Let x ∈ Γ and u ∈S|Ω. Let

Ix ≡{

i ∈ {1,2, · · · ,q} : x = hi(u′i)

for some u′i ∈U ′i}.

Then

γu(x) = ∑i∈Ix

(γH∗i (uψ i))(gi (x)) = ∑i∈Ix

(γH∗i (uψ i))(gi(hi(u′i)))

= ∑i∈Ix

(γH∗i (uψ i))(u′i).

Now because Hi is Lipschitz continuous and uψ ∈S, it follows that H∗i (uψ i) ∈ H1 (Rn)and is continuous and so by Theorem 38.5.7 on Page 1322 for a.e. u′i,

= ∑i∈Ix

H∗i (uψ i)(u′i,0

)= ∑

i∈Ix

h∗i (uψ i)(u′i)= ∑

i∈Ix

(uψ i)(hi(u′i))

= u(x) for µ a.e.x.

This verifies 38.6.26 and completes the proof of the theorem.