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similarly, we can prove
HÖLDER REGULARITY FOR ∂ 249
∂ Cdζ
∂zi
J
Aj+1 n
bD
∧ (zi − ζi)dζi
Bn−j−1 1
τ β ,
(z, ε)|z − ζ| 2n−2j−3
i=1
where β takes over all multiindices satisfying condition C. This completes
the proof of (1).
2) For z ∈ U i, ζ ∈ bD and ζ /∈ U ′ i , 〈∂ζr(ζ), ζ − z〉 = 0 and |z − ζ| = 0.
Note U1, · · · , UN covering bD. It follows
(4.24)
∂r
〈 (ζ), ζ − z〉
∂ζ 1, |z − ζ| 1
by compactness. It follows that the coefficients of differential forms dzA j,0
q
are bounded. The Proposition 4.1 is proved.
Now we are ready to prove Lemma 4.2.
Proof of Lemma 4.2. Define
(4.25)
S0 = {i; βi + β i = 0}
S1 = {i; βi + β i = 1}
S2 = {i; βi + β i = 2}
S3 = {i; βi + β i = 3}
for the multiindex β satisfying condition C. Denote the cardinal of Si by
ni, i = 0, 1, 2, 3. We know that n3 ≤ 1 from condition C. We consider three
cases: Case A, n3 = 0 and 1 ∈ S2; Case B, n3 = 0 and 1 /∈ S2; Case C,
n3 = 1.
Note
(4.26)
1 1
τl(z, ε) ε ≈
β
1
τ1(z, ε) .
If we replace τl by τ1 in (4.9) for some l ∈ S2, β1 + β 1 will increase 1. Case
B is reduced to Case A. In Case C, β1 + β 1 = 1. If we replace τl by τ1 for
l ∈ S3, β1 + β 1 will increase to 2. Case C is reduced to Case A. Thus we
only need to consider Case A.