46.2. THE TRACE ON THE BOUNDARY OF AN OPEN SET 1515

and

||u||2,hs,p,Γ ≡∑j,i

∣∣∣∣∣∣h∗i (uφ jψ i

)∣∣∣∣∣∣s,p,Ui

= ∑j,i

∣∣∣∣∣∣h∗i (uφ jψ i

)∣∣∣∣∣∣s,p,h−1

i

(Wi∩W ′j

)But from the assumptions on h and g, in particular the assumption that these are restrictionsof functions which are defined on open subsets of Rm which have Lipschitz derivativesup to order k along with their inverses, Theorem 45.4.4 implies, there exist constants Ci,independent of u such that∣∣∣∣∣∣h∗i (uφ jψ i

)∣∣∣∣∣∣s,p,h−1

i

(Wi∩W ′j

) ≤C1

∣∣∣∣∣∣g∗j (uφ jψ i

)∣∣∣∣∣∣s,p,g−1

j

(Wi∩W ′j

)and ∣∣∣∣∣∣g∗j (uφ jψ i

)∣∣∣∣∣∣s,p,g−1

j

(Wi∩W ′j

) ≤C2

∣∣∣∣∣∣h∗i (uφ jψ i

)∣∣∣∣∣∣s,p,h−1

i

(Wi∩W ′j

) .Therefore, the two norms, ||·||1,gs,p,Γ and ||·||2,hs,p,Γ are equivalent. It follows that the norms,

||·||2s,p,Γ and ||·||1s,p,Γ are equivalent. This proves the following theorem.

Theorem 46.1.3 Let Γ be described above. Then any two norms for W s,p (Γ) as in Defini-tion 38.6.3 are equivalent.

46.2 The Trace On The Boundary Of An Open SetNext is a generalization of earlier theorems about the loss of 1

p derivatives on the boundary.

Definition 46.2.1 Define

Rn−1k ≡ {x ∈ Rn : xk = 0} , x̂k ≡ (x1, · · · ,xk−1,0,xk+1, · · · ,xn) .

An open set, Ω is Cm,1 if there exist open sets, Wi, i = 0,1, · · · , l such that

Ω = ∪li=0Wi

with W0 ⊆Ω, open sets Ui ⊆Rn−1k for some k, and open intervals, (ai,bi) containing 0 such

that for i≥ 1,

∂Ω∩Wi = {x̂k +φ i (x̂k)ek : x̂k ∈Ui} ,

Ω∩Wi = {x̂k +(φ i (x̂k)+ xk)ek : (x̂k,xk) ∈Ui× Ii} ,

where φ i is Lipschitz with partial derivatives up to order m also Lipschitz. Here Ii = (ai,0)or (0,bi) . The case of (ai,0) is shown in the picture.