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HÖLDER REGULARITY FOR ∂ 243
Theorem 1.1 for κ ≥ 1 − 1
1
m obviously follows from κ < 1 − m . We will
prove (3.16) in Section 4.
We will need the following estimates.
Lemma 3.7. If a, b, a ′ , α, α ′ > 0, k ≥ 1, then
(3.17)
C
b 1
k
(a + b|ζ| k 1
α+ ) k
where dV (ζ) is the volume element of C.
Proof. Denote ζ = x + iy, then the integral
≤
b 1
k
dx ·
=
R
R
(a + b|x| k 1
α+ ) k
dx
(a + |x| k 1
α+ ) k
·
1
(a ′ + |ζ|) α′ 1
dV (ζ)
+1 aα 1
a ′α ′ ,
·
R
1
a α a ′α ′ .
Lemma 3.8. If a, b, α > 0, k ≥ 1, then
(3.18)
C
b 2
k
(a + b|ζ| k 2
α+ ) k
where dV (ζ) is the volume element of C.
Proof. Define
(3.19)
and
(3.20)
R
dy
(a ′ + |y|) α′ +1
dy
(a ′ + |y|) α′ +1
dV (ζ) 1
,
aα D1 = {ζ; b|ζ| k ≥ a}
D2 = {ζ; b|ζ| k < a}.
It follows that on the region D1, we have
b 2
k
dV (ζ)
D1
where L = ( a
b
(a + b|ζ| k 2
α+ ) k
D1
∞
L
b 2
k
(b|ζ| k 2
α+ ) k
dV (ζ)
b −α (ρ k 2
−
) k −α ρdρ a −α ,
) 1
k . On the region D2, we obtain the same upper bound
D2
b 2
k
(a + b|ζ| k 2
α+ ) k
dV (ζ)
b 2
k
a 2
k +α Vol(D2) a −α .
This completes the proof of Lemma 3.8.
Lemmas 3.7 and 3.8 were used in [MS] implicitly to estimate the Bergman
projection operator in the convex domain of finite type.