PEAK FUNCTION AND ITS APPLICATION
Sanghyun Cho †
Abstract. Let Ω be a smoothly bounded pseudoconvex domain in Cn and let bΩ
be of finite type and p ∈ bΩ. Assume that we have a precise optimal estimate of the
Bergman kernel and its derivatives near p. Then for each small neighborhood V of
p, there is a Hölder continuous peak function (at p) that extends holomorphically up
to bΩ r V .
1. Introduction.
Let Ω be a smoothly bounded pseudoconvex domain in Cn and let A(Ω) denote
the functions holomorphic on Ω and continuous on Ω. A point p ∈ bΩ is a peak point
if there is a function f ∈ A(Ω) such that f (p) = 1, and |f (z)| < 1 for z ∈ Ω − {p}.
The existence of peaking functions as well as the additional smoothness up to
the boundary is one of the major topics in several complex variables. When Ω
is strictly pseudoconvex, the situation with regard to peak functions is fairly well
understood, but in the weakly pseudoconvex case we know very little. If Ω ⊂⊂ C2
is pseudoconvex and bΩ is of finite type, Bedford and Fornaess [1], showed that
there is a peak function in A(Ω). This method also works for finite type domains in
Cn where the Levi-form of bΩ has (n-2)-positive eigenvalues. We also mention the
work of Bloom [2], Hakim and Sibony [11], and Range [16] on the existence of peak
functions with additional smoothness up to the boundary of Ω, i.e., in the various
subclass of A(Ω).
Recently the author proposed a method [8] to construct a peak function for
the domains in Cn where the optimal estimates of the Bergman kernel function
are known. Namely, for each neighborhood V of p ∈ bΩ we construct a regular
bumping family of pseudoconvex domains outside V , and use Bishop’s 41 − 34 method
on bumped domains. This is a modification of Fornaess and McNeal’s method
[10] and can be applied to wide class of domains in Cn . The optimal estimates
of the Bergman kernel function and its derivatives are known, for example, for
pseudoconvex domains of finite type in C2 [4,12], decoupled pseudoconvex domains
of finite type domains in Cn [13], pseudoconvex domains of finite type in Cn where
the Levi-form of bΩ has (n − 2)-positive eigenvalues [6,7], and for the locally convex
finite type domains in Cn [15]. In this paper, we want to prove the existence of the
Hölder continuous peak functions on the domains we have just mentioned.
2000 Mathematics Subject Classification. 32H10.
Key words and phrases. peak functions, Bergman kernel function, finite 1-type.
†Partially supported by Non directed research fund, K.R.F. 1994, and by GARC-KOSEF, 1995
Typeset by AMS-TEX
1
2
SANGHYUN CHO †
Theorem 1.1. Let Ω be a smoothly bounded pseudoconvex domain in Cn and let
bΩ be of finite type and p ∈ bΩ. Assume that optimal estimates of the Bergman
kernel function and its derivatives are known on and off the diagonal near p ∈ bΩ.
Then for each small neighborhood V of p, there is a Hölder continuous peak function
that peaks at p and extends holomorphically up to bΩ r V .
Remark 1.2. The author proved the existence of (continuous) peak functions for
the locally convex domains in Cn [8]. Here we prove an additional smoothness
(i.e., Hölder continuity) of the peak function up to the boundary for the domains,
for example, we have mentioned before Theorem 1.1.
Remark 1.3. Fornaess and McNeal also proved the existence of the Hölder continuous peak functions for the pseudoconvex domains of finite type in C2 and for
decoupled pseudoconvex finite type domains in Cn . Here we revise their proof and
we show, in addition, that the peak function extends holomorphically up to bΩ r V .
The existence of peak functions for A(Ω) implies that Ω is complete in the
Carathéodory metric. Since the Carathéodory metric is smaller than the Kobayashi
metric and the Bergman metric, we obtain the following corollary as an immediate
application of Theorem 1.1.
Corollary 1.4. Let Ω be one of four kinds of domains we mentioned before Theorem 1.1. Then Ω is complete in the Kobayashi, Bergman and Carathéodory metrics.
We will only prove the existence of Hölder continuous peak function on the
pseudoconvex domains of finite type in Cn where the Levi-form of bΩ has (n − 2)positive eigenvalues. The same analysis will give us a Hölder continuous peak
functions for the rest kinds of domains.
2. Smooth bumping families.
Let Ω be a smoothly bounded pseudoconvex domain in Cn with smooth defining
function r and let 0 ∈ bΩ. If g : C −→ C is any smooth function with g(0) = 0, let
ν(g) denote the order of vanishing of g at 0. For a vector valued G = (g1 , . . . gn ),
let ν(G) denote the minimum order of vanishing of the gi at 0.
Definition 2.1. (D’Angelo). 0 is a point of finite 1-type if
supG
ν(r ◦ G)
= ∆(0) < ∞,
ν(G)
where G : C −→ Cn is a complex analytic map with G(0) = 0; ∆(0) is called the
type of 0.
Definition 2.2. Let p ∈ bΩ be an arbitrary point and let V be a neighborhood of
p. By a smooth bumping family for Ω outside V we mean a family {Ωt }0≤t≤1 of
pseudoconvex domains with C ∞ defining functions {rt } with the following properties:
(a) Ω = Ω0 ,
(b) Ωt1 ⊂ Ωt2 if t1 < t2 , and rt (z) is smooth in z and t,
(c) for any neighborhood U of ∂ΩrV there is a t0 > 0 such that Dt rU = DrU
for all t ∈ [0, t0 ].
The following theorem can be found in [5].
PEAK FUNCTIONS IN Cn
3
Theorem 2.3. Let p be a point of finite 1-type in the boundary of a pseudoconvex
domain Ω in Cn with smooth defining function r(z). Then for each neighborhood V
of p, there exists a smooth 1-parameter family of pseudoconvex domains {Ωt }0≤t<t0 ,
each defined by Ωt = {z; r(z, t) < 0}, where r(z, t) has the following properties:
(a)
(b)
(c)
(d)
(e)
r(z, t) is smooth in z near bΩ, and in t for 0 ≤ t < t0 ,
r(z, t) = r(z) for z ∈
/ V,
∂r
∂t (z, t) ≤ 0,
r(z, 0) = r(z),
for z in V , ∂r
∂t < 0.
Definition 2.4. Suppose Ω, p ∈ bΩ, V be as Theorem 2.3. Then we say {Ωt }0≤t<t0
a bumping family of Ω with front V .
Theorem 2.5. Let Ω ⊂⊂ Cn be a smoothly bounded pseudoconvex domain and let
bΩ be of finite 1-type. Assume p ∈ bΩ and V is a small neighborhood of p. Then
there is a 1-parameter family of a smooth bumping family {Ωt } outside V .
Proof. Choose a neighborhood U of p such that V ⊂⊂ U . Since bΩ is compact, we
can choose points z1 , . . . , zN ∈ bΩ and ²1 , . . . , ²N > 0 such that
(1)
(2)
(3)
(4)
Ω is pseudoconvex and bΩ is of finite type,
∪N
i=1 B(zi , ²i /2) ⊃ bΩ r U ,
V ∩ B(zi , ²i ) = ∅, i = 1, 2, . . . , N ,
B(zi , ²i ), is contained in a neighborhood Vi where Theorem 2.3 can be
applicable, i = 1, 2, . . . , N .
Set Vi = B(zi , ²i ), i = 1, 2, . . . , N , for the convenience. Consider a bumping family
of Ω with front V1 . Since the type condition is stable under small C ∞ -perturbations
of bΩ, we will get a family {Ωt1 }0≤t1 <α1 of smooth pseudoconvex domains satisfying
(1)–(4) for the domains Ωt1 (instead of Ω) provided α1 is sufficiently small. For
each Ωt1 , 0 ≤ t1 < α1 , we consider a bumping family of Ωt1 with front V2 and
call it {Ωt1 t2 }0≤t2 <α2 . Again {Ωt1 t2 }0≤t2 <α2 will satisfy (1)–(4) provided α2 is
sufficiently small. Continuing in this manner, we will get a bumping family of
pseudoconvex domains {Ωt1 t2 ...tN } outside V . Obviously we can regard this family
as a 1-parameter family of pseudoconvex domains. ¤
3. Estimates on the Bergman kernel.
Let Ω be a bounded pseudoconvex domain of finite type in Cn with smooth
defining function r and let p ∈ bΩ. Suppose that p is a point of finite T in the
sense of D’Angelo [9], and assume that the Levi-form of bΩ has (n − 2)-positive
eigenvalues at p. In this section we estimate the Bergman kernel function near p
using the analysis of local geometry of bΩ in [6,7]. For the estimates of the kernel
on the other domains, one can refer [4,12,13,15].
Let us take the coordinate functions z1 , . . . , zn about p ∈ bΩ. After a linear
∂r
change of coordinates, we may assume that | ∂z
(z)| ≥ c > 0 for all z ∈ U , for some
1
neighborhood of p.
Proposition 3.1. ([4, Proposition 2.2 ]) For each z 0 ∈ U and positive even integer
−1
0
m, there is a biholomorphism Φz0 : Cn −→ Cn , Φ−1
z 0 (z ) = 0, Φz 0 (z) = (ζ1 , . . . , ζn )
4
SANGHYUN CHO †
such that
(3.1)
r(Φz0 (ζ)) = r(z 0 ) + Reζ1 +
n−1
X
³
´
0 j k
Re bα
(z
)ζ
ζ
ζ
j,k
n n α
X
α=2 j+k≤ m
2
j,k>0
X
+
aj,k (z
0
k
)ζnj ζ n
+
n−1
X
|ζα |2
α=2
j+k≤m
j,k>0
m
+ O(|ζ1 ||ζ| + |ζ 00 |2 |ζ| + |ζ 00 ||ζn | 2 +1 + |ζn |m+1 ).
Let us choose z 0 near p and fix it for a moment. We take m is equal to T in
Proposition 3.1. Set ρ(ζ) = r ◦ Φz0 (ζ), and set
τ (z 0 , δ) = min{(
1
δ
l : 2 ≤ l ≤ T }.
)
Al (z 0 )
Since AT (z0 ) ≥ c > 0, it follows that AT (z 0 ) ≥ c0 > 0 for all z 0 ∈ U if U is
sufficiently small. This gives the inequality,
1
1
δ 2 . τ (z 0 , δ) . δ T , z 0 ∈ U.
(3.2)
The definition of τ (z 0 , δ) easily implies that if δ 0 < δ 00 , then
(3.3)
1
1
(δ 0 /δ 00 ) 2 τ (z 0 , δ 00 ) ≤ τ (z 0 , δ 0 ) ≤ (δ 0 /δ 00 ) T τ (z 0 , δ 00 ).
1
Now set τ1 = δ, τ2 = . . . = τn−1 = δ 2 , τn = τ (z 0 , δ) = τ and define
(3.4)
Rδ (z 0 ) = {ζ ∈ Cn ; |ζk | < τk , k = 1, 2, . . . , n}, and
Qδ (z 0 ) = {Φz0 (ζ); ζ ∈ Rδ (z 0 )}.
In [1], the author proved that for z ∈ Qδ (z 0 ) it follows that
τ (z 0 , δ) ≈ τ (z, δ).
(3.5)
We recall the estimates on the Bergman kernel function and its derivatives for the
domain Ω considered in this section [6,7]. Because of the transformation formula
for the Bergman kernel function, we state our estimates the special coordinates in
Proposition 3.1.
Theorem 3.2. Let Ω and p ∈ bΩ be as in the begining of this section. For z 1 , z 2 ∈
i
U ∩ Ω, set ζ i = Φ−1
z 1 (z ), i = 1, 2. Then there exist a neighborhood U of p and
constants Cα,β , independent of z 1 , z 2 ∈ U ∩ Ω, such that
β
1
|Dζα1 Dζ 2 KΩz1 (ζ 1 , ζ 2 )| ≤ Cα,β δ −n−α1 −β1 − 2 γ · τ (z 1 , δ)−2−αn −βn
where δ = |ρ(ζ 1 )| + |ρ(ζ 2 )| + M (ζ 1 , ζ 2 ), γ = α2 + β2 + . . . + αn−1 + βn−1 , and
Pn−1
PT
M (ζ 1 , ζ 2 ) = |ζ11 −ζ12 |+ j=2 |ζj1 −ζj2 |2 + l=2 Al (z 0 )|ζn1 −ζn2 |l , and where ρ = r◦Φz1 .
PEAK FUNCTIONS IN Cn
5
Remark 3.3. Let Ω and p ∈ bΩ be as above. Then the author [6] estimated the
Bergman kernel function K(z, z) on the diagonal as follows
(3.6)
K(z, z) ≈
m
X
2
2
Al (z 0 ) l r(z)−n− l ≈ r(z)−n · τ (z, r(z))−2 ,
l=2
near p, and this is the case when z 1 = z 2 , and α, β = 0, in Theorem 3.2.
Since p is a point of finite type T , Catlin’s theorem [3] says that there are ² > 0
and a neighborhood U in which ∂-Neumann problem satisfies a subelliptic estimate
of order ² > 0 on (0,1)-forms. For each w ∈ U ∩ Ω, define the function
hw (z) =
KΩ (z, w)
.
KΩ (w, w)
Then the estimates of Theorem 3.2 and (3.6) give us
(3.7)
1
|Dzα hw (z)| . δ −α1 − 2 (α2 +... ,αn−1 ) · τ (z, δ)−αn ,
where δ = |r(w)| + |r(z)| + M (w, z), and where
M (w, z) = |z1 − w1 | +
n−1
X
|zj − wj |2 +
j=2
T
X
Al (w)|zn − wn |l .
l=2
We will estimate KΩ (z, w) for z outside a certain neighborhood of w and will
show that |hw (z)| is quite small outside that neighborhood. For the convenience in
notation, we denote by C the various constants that follows. Let π be a projection
onto bΩ and set p = π(w). For η > 0, let B(p, η) denote the euclidean ball centered
at p of radius η. Let ζ ∈ C ∞ (Cn ) be a function with the property that ζ ≡ 1 on
Ω r B(p, η) and ζ ≡ 0 on B(p, η2 ), and let N denote the ∂-Neumann operator on
(0,1)-forms. The following theorem was proved in [14].
Proposition 3.4. Let Ω, U , and ² > 0 be as above. Let s, t ∈ R+ . If α is a smooth
(0,1)-form in the domain of the Kohn Laplacian and suppα ⊂ B(p, η8 ) then there
is a constant Cst > 0 so that
kζN αk2s ≤ Cst η −2(
s+t
² +4)
kαk2−t .
Let φ ∈ C0∞ (0, 1) be a non-negative radial function with
set
δ(w) −2n z − w
φw (z) = (
)
φ(
).
2
δ(w)/2
R
φ = 1. For w ∈ U ,
Then from the Kohn’s formula,
∗
KΩ (z, w) = φw (z) − ∂ N ∂φw (z).
Assume suppφw ⊂ B(p, η8 ). Then from the Proposition 4.1 with s = r + 1, we have
kζ(·)KΩ (·, w)k2r ≤ CkζN ∂φw (·)k2r+1
≤ Cη −2(
r+1+t
+4)
²
k∂φw (·)k2−t .
6
SANGHYUN CHO †
If t > n + 1, Sobolev’s lemma gives
kφw k2−(t−1) = sup{|(φw , f )|; f ∈ C0∞ , kf kt−1 ≤ 1}
Z
≤ C| φw | ≤ C
for some C > 0. If we choose r > n + 1, another application of Sobolev’s lemma
shows
supz |ζ(z)KΩ (z, w)| ≤ Ckζ(·)KΩ (·, w)kr .
Hence
supz |ζ(z)KΩ (z, w)| ≤ Cη −(
2n+4
+4)
²
.
²
Now set η = δ(w) 2(2n+4) . If we combine Theorem 3.2 and the estimate (3.6), we
have proved the following
Proposition 3.5. Let Ω and η be as above. There exists a constant C > 0, independent of w ∈ U , so that
|hw (z)| ≤ Cδ(w)
for w ∈ U and z ∈ Ω r B(π(w), η).
Remark 3.6. In [1], the author showed that the sharp subelliptic estimate holds
near p. So we may, in fact, take ² = T1 .
4. A construction of Hölder continuous peak functions.
In this section, we construct a Hölder continuous peak function at p. This
construction can be done by careful observation of the estimates in Theorem 3.2
and (3.6), (3.7). Let Ω and p ∈ bΩ be as in the begining of section 3 and assume
that the type of p is T . Then by Remark 3.6, the sharp subelliptic estimate of order
1
1
T holds near p. Set η = 2(2n+4)T . Now we denote by N the interior normal to the
boundary of Ω at p.
Lemma 4.1. For every q on N , let |q − p| = d. There exists a constant C > 0
such that for every point q on N sufficiently close to p, there exist a neighborhood
Up of p and a holomorphic function h = hq on Ω such that
(1)
(2)
(3)
(4)
|h| < C on Ω,
h(q) = 1,
|h(z)| < Cd for z ∈ Ω r Up ,
|Dh| < Cd .
Proof. Define hq (z) = K(z, q)/K(q, q). Property (2) is clear. From Proposition 3.5,
we have |hq (z)| ≤ Cd for z ∈ Ω r B(π(q), dη ). This proves (3) with Up = B(p, dη ).
The estimates of Theorem 3.2 and (3.6) give |h| ≤ C for z ∈ Up = B(π(q), dη ).
This fact together (3) gives (1). Also from the estimates of Theorem 3.2 and (3.6),
(3.7), we have |Dh| < Cd . ¤
PEAK FUNCTIONS IN Cn
7
Remark 4.2. Actually (4) of Lemma 4.1 can be sharpened as in (3.7).
We now ready to costruct a Hölder continuous peak functions at p;
For simplicity we assume that p = 0. Let the type of p is equal to T and choose
a neighborhood U of p such that the subelliptic estimates for ∂-equation of order T1
hold on U and Theorem 3.2 and the estimates of Remark 3.3 hold on U . We denote
by N the interior normal to the boundary of Ω at p. For each neighborhood V ⊂⊂ U
of p, choose a neighborhood V1 , V1 ⊂⊂ V ⊂⊂ U . Next we consider a 1-parameter
family of a smooth bumping family {Ωt }0≤t<t0 outside V1 . We may assume that
V1 ∩ Ω = V1 ∩ Ωt for all t > 0, after perhaps shrinking V1 , and Ω r V ⊂⊂ Ωt r V
for all 0 < t < t0 . Now fix 0 < t1 < t0 and consider the pseudoconvex domain Ωt1 .
Since the type condition is stable, we may assume that bΩt1 is of finite type. Also
s
we have d = dist (Ω r V, bΩt1 r V ) > 0. Let qn = p + 2nT
N , for a small constant
s
s to be determined. Hence dn = 2nT . Define hn (z) = hqn (z + qn − p), where hqn
is the function defined on Ωt1 (instead of Ω) as in Lemma 4.1 associated with qn .
Notice that hn (z) is defined on Ω r V , and on U intersect a translate of Ω which
contains Ω ∩ V , provided s is sufficiently small. Therefore hn is well defined and
holomorphic on Ω for each n ≥ 0. Let Un denote the neighborhood corresponding
c < 1, to be determined later,
to hn as in Lemma 4.1. For a suitable constant 0 < P
∞
let r = 1 − c and define a peak function as H = r n=0 cn hn . During the proof
we will choose the constants more precisely. Let us estimate H on various sets.
First outside U0 . By the property (3) of Lemma 4.1 we have |hn | < 21 for every n
P∞ n
provided that s is sufficiently small. So we get that |H| < r n=0 c2 = 12 .
PT
Pn−1
Now let P(z) = |z1 | + j=2 |zj |2 + l=2 Al (0)|zn |l , with the notation of section
3. Next assume that z is in Un r Un+1 . Let m be the largest integer so that
P(z) < dm . Then from the estimates of the first derivatives of hk and by virtue of
(3.2) and (3.5), we have for k ≤ m that
|hk | − 1 ≤
C[d−1
k |z1 |
+
n−1
X
−1
dk 2 |zj | + τ (0, dk )−1 |zn |]
j=2
≤ C[d−1
k |z1 | +
n−1
X
−1
dk 2 |zj | + τ (0, dk )−1 · τ (0, P(z))]
j=2
because τ (0, P(z))−1 ≤
C
|zn | .
Thus by virtue of (3.3) and (3.4) we get that
1
− 21
2
|hk | − 1 ≤ C[dm · d−1
k + dm · dk
+ τ (0, dk )−1 · τ (0, dm )]
T
≤ C(2T (k−m) + 2 2 (k−m) + 2k−m ).
Therefore |hk | < 1 + C2k−m for k ≤ m. Similarly, if we combine the estimates in
d2k
Theorem 3.2 and (3.6), we have that |hk | < C P2 (z)
for m < k < n. Also, |hn | < C
8
SANGHYUN CHO †
s
and |hk | < C 2kT
if k > n. Hence we estimate H as follows:
|H| < r[
X
ck (1 + C2k−m ) +
k≤m
X
ck C
m<k<n
m+1
X
dk
s
n
k
+
c
C
+
c
C
]
P2
2k
k>n
m+1
(1 − c
)
((2c)
− 1)
+ rC
(1 − c)
(2c − 1)2m
( 1 )n+1
4
+ rCcm+1 · + rcn C + rCs 2
3
(1 − c)
2C(1 − c) m+1
= (1 − cm+1 ) +
(c
− 2−m−1 )
(2c − 1)
4
1
+ rCcm+1 + rcn C + Cs( )n+1 .
3
2
<r
Now set r =
1
10C ,
(i.e., c = 1 −
1
10C )
for instance. Then
1
(cm+1 − 2−m−1 )
5(2c − 1)
2
1
1
+ cm+1 + cn + Cs( )n+1
15
10
2
1 m+1 1 1 n+1
<1− c
+ ( )
< 1,
2
4 2
|H| < (1 − cm+1 ) +
9
if s is sufficiently small. Here we assumed that C >> 1 and hence 1 > c > 10
, for
instance.
Next we consider Hölder estimates. Let Vk = {z : |z| ≤ 2−kT } and choose
any z, w ∈ V1 . Without loss of generality,
we may assume that 0 < |w| < |z|.
P
We will estimate |H(z) − H(w)| < r cn |hn (z) − hn (w)|. First fix m ≤ n so
that w ∈ Vn r Vn+1 and z ∈ Vm r Vm+1 . From (4) of Lemma 4.1, we have
|hj (z) − hj (w)| ≤ C|z − w| · dj−1 ≤ C|z − w|2jT . Without loss of generality, we may
assume that mlog 1c ≥ 2(10T + 1)log2. We thus get that
X
r
cj |hj (z) − hj (w)| ≤ C|z − w| · (2T c)m+10
j<m+10
≤ C|z − w|ν ,
if |z − w|1−ν · (2T c)m+10 ≤ 1, which holds if ν <
We consider next the tail end of the series.
log 1c
2T log2 .
Case I. n > m + 3
Then we have
r
X
cj |hj (z) − hj (w)| ≤ Ccm .
j>m+8
We need cm ≤ |z − w|ν or cm ≤ 2−mT ν . So if we take ν ≤
r
X
j>m+8
log 1c
T log2 ,
cj |hj (z) − hj (w)| ≤ C|z − w|ν .
then
PEAK FUNCTIONS IN Cn
9
Case II. n ≤ m + 3 and |z − w| ≤ C|w|T .
Note that |hj (z) − hj (w)| ≤ C|Dhj (z 0 )||z − w| for some z 0 ∈ QM (z,w) (z). Since
|w| ≈ |z 0 | ≈ 2−mT , the estimate (3.7) gives us
|hj (z) − hj (w)| ≤ C2mT |z − w|
1
≤ C2mT |z − w| T · |w|T −1
1
≤ C|z − w| T .
So
r
X
1
cj |hj (z) − hj (w)| ≤ C|z − w| T .
j>m+8
Case III. n ≤ m + 3 and |z − w| >> |w|T .
Let z̃, w̃ be the projections of z and w onto bΩ respectively and let us denote
νz̃ , νw̃ the corresponding outward normal vectors at z̃ and w̃. Set z 0 = z − |w|νz̃
and w0 = w − |w|νw̃ . Note that
|hj (z 0 ) − hj (w0 )| ≤ C|Dhj (z 00 )||z 0 − w0 |,
where |r(z 00 )| & |w|. Again from the estimate in (3.7), we get that
|hj (z 0 ) − hj (w0 )| ≤ C|w|−1 |z 0 − w0 | ≤ C2mT |z 0 − w0 | . 1.
Therefore,
r
X
cj |hj (z) − hj (w)| ≤ r
j>m+8
X
cj |hj (z) − hj (z 0 )|
j>m+8
+r
X
cj |hj (z 0 ) − hj (w0 )|
j>m+8
+r
X
cj |hj (w0 ) − h(w)|
j>m+8
m
≤ C(c + cm + cm ) ≤ Ccm .
2
We want cm ≤ |z − w|ν , which will follow if cm ≤ |w|T ν ≈ 2−mT ν . This leads
P∞
log 1c
to the estimate ν < T 2 log2
. So H = r n=0 cn hn is a Hölder continuous (of order
ν≤
log 1c
T 2 log2 )
peak function at p.
References
1. Bedford, E. and Fornaess, J.E., A construction of peak functions on weakly pseudoconvex
domains, Ann. of Math. 107 (1978), 555–568.
2. Bloom, T., C ∞ peak functions for pseudoconvex domains of strict type, Duke Math. J. 45
(1978), 133–147.
3. Catlin, D.W., Subelliptic estimates for the ∂-Neumann problem on pseudoconvex domains,
Ann. of Math. 126 (1987), 131–191.
4. Catlin, D.W., Estimates of invariant metrics on pseudoconvex domains of dimension two,
Math. Z. 200 (1989), 429–466.
10
SANGHYUN CHO †
5. Cho, S., Extension of complex structures on weakly pseudoconvex compact complex manifolds
with boundary, Math. Z. 211 (1992), 105–120.
6. Cho, S., Boundary behavior of the Bergman kernel function on some pseudoconvex domains
in Cn , Trans. of A.M.S. 345 (1994), 803–817.
7. Cho, S., Estimates of the Bergman kernel function on certain pseudoconvex domains in Cn ,
Math. Z. (to appear).
8. Cho, S., A construction of peak functions on locally convex domains in Cn , Nagoya Math. J.
140 (1995), (to appear).
9. D’Angelo, J., Real hypersurfaces, order of contact, and applications, Ann. of Math. 115
(1982), 615–637.
10. Fornaess, J.E. and McNeal, J., A construction of peak functions on some finite type domains,
Amer. J. of Math. 116 (No. 3) (1994), 737–755.
11. Hakim, M. and Sibony, N., Quelques conditions pour l’existence de fonctions pics dans des
domaines pseudoconvexes, Duke Math. J. 44 (1977), 399–406.
12. McNeal, J., Boundary behavior of the Bergman kernel functions in C2 , Duke Math. J. 58
No. 2 (1989), 499–512.
13. McNeal, J., Local geometry of Decoupled pseudoconvex domain, Proc. in honor of Grauert,
H., Aspekte der Math., Vieweg, Berlin (1990), 223–230.
14. McNeal, J., Lower bounds on the Bergman metric near a point of finite type, Ann. of Math.
136 (1992), 361–373.
15. McNeal, J., Estimates on the Bergman kernels of convex domains, Adv. in Math. (to appear).
16. Range, R.M., The Caratheodory metric and holomorphic maps on a class of weakly pseudoconvex domains, Pacific J. Math. 78 (1978), 173–188.
Department of Mathematics Education, Pusan University, Pusan 609–735, KOREA
E-mail address: [email protected]