For printing - MSP

Note
(4.7)
bD∩U ′ i
dzA j
J
HÖLDER REGULARITY FOR ∂ 245
(z, ζ) ∧ f
bD∩U ′ i
fL ∞ 0,q
dzA j
J (z, ζ) ∧ fbDdV (ζ)
bD∩U ′ i
where dV (ζ) is the volume element of bD, and
dzA j
JbD (4.8)
1
bD\U ′ i
dzA j
J (z, ζ)bDdV (ζ),
by (4.6) for z ∈ Ui, and |r(z)| ≈ δD(z), the proof of Proposition 3.6 is
reduced to the following estimate.
Lemma 4.2. For β satisfying condition C and z ∈ Ui for some i, we have
(4.9)
Iβ =
bD∩U ′ i
1
τ β 1
dV (ζ) |r(z)|−1−κ+ m , ε = d(z, ζ)
(z, ε)|z − ζ| 2n−2j−3
for 0 < κ < 1 − 1
m , where dV (ζ) is the volume element of bD.
Before we begin to prove Proposition 4.1, we give a lemma. Since
n
(4.10)
∂ζ∂ζβ = dζi ∧ dζi,
we get
(4.11) A j,0
q−1 =
i=1
∂z∂ζβ = −
∂zβ =
n
dzi ∧ dζi,
i=1
n
(zi − ζi)dzi,
i=1
1
Aj+1βn−j−1 ∂ζr ∧ (∂ζ∂ζr) j ∧ ∂ζβ
n
n−q−3−j
n
q−1 ∧ dζi ∧ dζi ∧ dζi ∧ dzi ,
i=1
where A = 〈∂ζr(ζ), ζ − z〉. Set
(4.12)
C = ∂ζr ∧ (∂ζ∂ζr) j .
Lemma 4.3. For z ∈ Ui, ζ ∈ bD ∩ U ′ i for some i, we have
CbD
(4.13)
L
i=1
ε j+1
τ L (z, ε) , dzC = 0, ε = d(z, ζ),