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246 WEI WANG
where the sum takes over all multiindices satisfiying
0 ≤ Li, Li ≤ 1,
n
i=1
Li + Li = 2j + 1, L1 = 0.
Proof of Lemma 4.3. dzC = 0 is obvious since C does not depend on z.
Now fix z ∈ Ui. Note formula (3.11) for A j,0
q is stated in the standard coordinates
ζ1, · · · , ζn in Cn . Denote the d(z, ζ)-extremal coordinates centered
at z by w1, · · · , wn. Then there exists an unitary matrix Uz, which is only
depending on z, and the translation Tz from the origin to z, such that Uz ◦Tz
transforms coordinates ζ1, · · · , ζn to coordinates w1, · · · , wn. It follows from
the invariance of differential forms under a linear transform that we can
write ∂ζr, ∂ζ∂ζr in coordinates w1, · · · , wn as
(4.14)
Thus
(4.15)
C = ∂wr ∧ (∂w∂wr) j
=
l1,... ,lj
t,k1,... ,kj
∂r
·
∂wt
∂ζr = ∂wr, ∂ζ∂ζr = ∂w∂wr.
j
i=1
∂ 2 r
∂wli ∂wki
· dwt ∧ dwl1 ∧ dwk1 ∧ · · · ∧ dwlj ∧ dwkj ,
where l1 · · · , lj are different, and t, k1, · · · , kj are different. Notice dr = 0
when restricted to the space tangential to bD, we find that
(4.16)
∂r
∂w1
dw1 = − ∂r
dw2 − · · · −
∂w2
∂r
dwn
∂wn
− ∂r
dw1 −
∂w1
∂r
dw2 − · · · −
∂w2
∂r
dwn
∂wn
holds on tangential space T (bD), dw1 disappeared in the differential forms
in the right side of (4.15) if we substitute (4.16) into (4.15) (see [CKM,