38.6. SOBOLEV SPACES ON MANIFOLDS 1327

≤C

(s

∑j=1

∣∣∣∣∣∣h′∗j (ψ′ju)∣∣∣∣∣∣2

Ht(

U ′j))1/2

= ||u||Ht (Γk).

where Lemma 38.3.3 on page 1312 was used in the last step. Now also, from Lemma38.3.3 on page 1312

||u||Ht (Γk)=

(s

∑j=1

∣∣∣∣∣∣h′∗j (ψ′ju)∣∣∣∣∣∣2

Ht(

U ′j))1/2

=

(s

∑j=1

∣∣∣∣∣∣(gk ◦h′j)∗h∗k

(ψ′ju)∣∣∣∣∣∣2

Ht(

U ′j))1/2

≤C

(s

∑j=1

∣∣∣∣∣∣h∗k (ψ′ju)∣∣∣∣∣∣2

Ht (Uk)

)1/2

≤C

(s

∑j=1||h∗ku||2Ht (Uk)

)1/2

=Cs ||h∗ku||Ht (Uk)= |||u|||t .

This proves the lemma.

38.6.2 The Trace On The Boundary

Definition 38.6.7 A bounded open subset, Ω, of Rn has a Cm,1boundary if it satisfies thefollowing conditions. For each p∈ Γ≡Ω\Ω, there exists an open set, W , containing p, anopen interval (0,b), a bounded open box U ′ ⊆Rn−1, and an affine orthogonal transforma-tion, RW consisting of a distance preserving linear transformation followed by a translationsuch that

RWW =U ′× (0,b), (38.6.23)

RW (W ∩Ω) = {u ∈ Rn : u′ ∈U ′, 0 < un < φW(u′)} ≡UW (38.6.24)

where φW ∈ Cm,1(U ′)

meaning φW is the restriction to U ′ of a function, still denoted byφW which is in Cm,1

(Rn−1

)and

inf{

φW(u′)

: u′ ∈U ′}> 0

The following picture depicts the situation.

R

W

Ω⋂

W RW (Ω⋂

W )

0

b

u′ ∈U ′

For the situation described in the above definition, let hW : U ′→ Γ∩W be defined by

hW(u′)≡ R−1

W(u′,φW

(u′))

, gW (x)≡ (RW x)′ , HW (u)≡ R−1W(u′,φW

(u′)−un

).

where x′ ≡ (x1, · · · ,xn−1) for x = (x1, · · · ,xn). Thus gW ◦hW = id on U ′ and hW ◦gW = idon Γ∩W. Also note that HW is defined on all of Rn is Cm,1, and has an inverse with the