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HÖLDER REGULARITY FOR ∂ 251
For l ∈ S2, in the 2 q−1 |r(z)|-extremal coordinates centered at z,
(4.31)
I q
β,i2···in
Πj=l[A j
ij (z)]
βj +β j
ij R2n−3
[A
C
l ⎛
2
i
il (z)] l ⎝2 q |r(z)| +
⎞
|ζk| ⎠
k=l
· 2 q |r(z)| + |ζ1| + m
A k t (z)|ζk| t
+ A l il
il (z)|ζl|
k=l t=2
−2−···− 2
−··· il βn+βn in
−κ
dx2 · · · dx2n
−(2n−2j−3+κ)
by the volume element dV (ζ) on bD ≈ dx2 · · · dxn. Now apply Lemma 3.8
to (4.31) with
to get
(4.32)
b = A l il (z), k = il, α = 2 +
a = 2 q |r(z)| + |ζ1| +
I q
j
Πj=l[A
β,i2···in ij (z)]
βj +β j
ij R2n−3 ⎛
· ⎝2 q |r(z)| + |ζ1| + m
k=l t=2
m
k=l t=2
j=l,j≥2
A k t (z)|ζk| t
βj + β j
ij
⎛
⎝2 q |r(z)| +
⎞
|ζk| ⎠
A k t (z)|ζk| t
k=l
⎞
⎠
−2− P
j=l,j≥2
+ κ
−(2n−2j−3+κ)
β j +β j
i j
−κ
dV (ζ)