Families of strictly pseudoconvex domains and peak functions

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by  Arkadiusz Lewandowski

FAMILIES OF STRICTLY PSEUDOCONVEX DOMAINS
AND PEAK FUNCTIONS
7
1
ARKADIUSZLEWANDOWSKI
0
2
n Abstract. Weprovethatgivenafamily(Gt)ofstrictlypseudoconvex
a domainsvaryinginC2 topology ondomains,thereexistsacontinuously
J varying family of peak functions ht,ζ for all Gt at every ζ ∈∂Gt.

6
1

V] 1. Introduction
C Let D ⊂ Cn be a bounded domain and let ζ be a boundary point of D.
h. It is called a peak point with respect to O(D), the family of functions which
t are holomorphic in a neighborhood of D, if there exist a function f ∈ O(D)
a
m such that f(ζ)= 1 and f(D\{ζ}) ⊂ D := {z ∈ C :|z| < 1}. Such a function
is a peak function for D at ζ. The concept of peak functions appears to
[
be a powerful tool in complex analysis with many applications. It has been
1
used to show the existence of (complete) proper holomorphic embeddings
v
9 of strictly pseudoconvex domains into the unit ball BN with large N (see
5 [5],[3]), to estimate the boundary behavior of Carathéodory and Kobayashi
2
metrics ([1],[6]), or to construct the solution operators for ∂ problem with
4
0 L∞ or Hölder estimates ([4],[10]), just to name a few of those applications.
. It is well known that every boundary point of strictly pseudoconvex do-
1
0 main is a peak point. Even more is true, in [6] it is showed that, given a
7
strictly pseudoconvex domain G, there exists an open neighborhood G of
:1 G, and a continuous function h : G×∂G → C such that for ζ ∈ ∂G,“the
v “
function h(·;ζ) is a peak function for G at ζ.
i
X In a recent paper [2] the following question has been posed:
ar Problem 1.1. Let ρ: D×Cn → R be a plurisubharmonic function of class
C2+k,k ∈ N∪{0}, such that for any z ∈ D the truncated function ρ|{z}×Cn
is strictly plurisubharmonic. Define G := {w ∈ Cn : ρ(z,w) < 0},z ∈ D.
z
This can be understood as a family of strictly pseudoconvex domains over
D. Does there exist a Ck-continuously varying family (hz,ζ)z∈D,ζ∈∂Gz of peak
functions for G at ζ?
z
We answer this question affirmatively in the case k = 0 and under addi-
tional assumption that, rougly speaking, the function ρ keeps its regularity
up to the set Ω×Cn, where Ω is some open neighborhood of D. Namely, let
us consider the following:
2010 Mathematics Subject Classification. Primary 32T40; Secondary 32T15.
Key words and phrases. strictly pseudoconvex domains, peak functions.
1
2 ARKADIUSZLEWANDOWSKI
Situation 1.2. Let (G ) be a family of bounded strictly pseudoconvex
t t∈T
domains,whereT ⊂ Cisacompactset. SupposewehaveadomainU ⊂⊂ Cn
such that
(1) ∂G ⊂⊂U,
[ t
t∈T
(2) for each t ∈ T there exists a defining function r for G satisfying
t t
with neighborhood ∂G ⊂ U all the conditions (A)-(F) below (see
t
Section 2),
(3) for any ε > 0 there exists a δ > 0 such that for any s,t ∈ T with
|s−t| ≤ δ there is krt−rskC2(U) < ε.
Observe that the above setting is completely in the spirit of the formula-
tion of Problem 1.1:
(i) The assumption that all the functions r satisfy (A)-(F) with common
t
neighborhood ∂G ⊂ U stays in relation with the fact that in Problem
t
1.1 all the defining functions for domains D have the same domain of
z
definition (Cn).
(ii) The assumption(3) comes fromthefactthatthefunction ρinProblem
1.1 is of class at least C2.
(iii) The compactness of the set of parameters (T) reflects the above men-
tioned assumption that ρcontinues to beof class C2 up toΩ×Cn, with
Ω being some neighborhood of D.
We shall prove the following:
Theorem 1.3. Let (G ) be a family of strictly pseudoconvex domains as
t t∈T
in Situation 1.2. Then there exists an ε > 0 such that for any η < ε there
1
exist an η > 0 and positive constants d ,d such that for any t ∈ T there
2 1 2
exist a domain G containing G , and functions h (·;ζ) ∈ O(G ),ζ ∈ ∂G
t t t t t
fulfilling the follocwing conditions: c
(a) h (ζ;ζ) = 1,|h (·;ζ)| < 1 on G \{ζ} (in particular, h (·;ζ) is a peak
t t t t
function for G at ζ),
t
(b) |1−h (z;ζ)| ≤ d kz−ζk,z ∈ G ∩B(ζ,η ),
t 1 t 2
(c) |h (z;ζ)| ≤ d < 1,z ∈ G ,kz −cζk≥ η .
t 2 t 1
Moreover, the constants ε,η ,d ,d , domains G , and functions h (·;ζ) may
2 1 2 t t
be chosen in such a way that for any α > 0 acnd any fixed triple (t ,ζ ,z ),
0 0 0
where t ∈ T,ζ ∈ ∂G , and z ∈ G , there exists a δ > 0 such that
0 0 t0 0 t0
whenever the triple (s,ξ,w) satisfies sd∈ T,ξ ∈ ∂G ,w ∈ G , and max{|s−
s s
”
t |,kξ −ζ k,kw−z k} < δ, then |h (z ;ζ )−h (w;ξ)| < α.
0 0 0 t0 0 0 s
The latter property will be referred to as continuity.
Remark 1.4. It is known that for each t ∈ T there exists an ε = ε(t) > 0
such that for any η < ε there exist a positive η = η (t) < η , constants
1 2 2 1
d = d (t),d = d (t) ∈ R,domainG containing G ,andfunctionsh (·;ζ) ∈
1 1 2 2 t t t
O(G ),ζ ∈ ∂G satisfying (a)-(c). Tfhis is a subject of Theorem 19.1.2 from
t t
c
FAMILIES OF STRICTLY PSEUDOCONVEX DOMAINS AND PEAK FUNCTIONS 3
[7]. The strength of our result dwells in the fact that all the constants
ε,η ,d ,d are chosen independently of t and in the continuity property.
2 1 2
In Section 2 we recall some preliminaries concerning the strictly pseudo-
convex domains. The proof of Theorem 1.3 is presented in Section 3.
2. Strictly pseudoconvex domains
Let D ⊂⊂ Cn be a domain. It is called a strictly pseudoconvex if there
exist a neighborhood U of ∂D and a defining function r : U → R of class C2
and such that
(A) D∩U = {z ∈ U :r(z) < 0},
(B) (Cn \D)∩U = {z ∈U :r(z) > 0},
(C) ∇r(z) 6= 0 for z ∈ ∂D, where ∇r(z):= Ä∂∂zr1(z),··· , ∂∂zrn(z)ä,
together with
C
L (z;X) > 0 for z ∈∂D and nonzero X ∈T (∂D),
r z
C
where L denotes the Levi form of r and T (∂D) is the complex tangent
r z
space to ∂D at z.
ItisknownthatU andrcanbechosentosatisfy(A)-(C)and,additionally:
(D) L (z;X) > 0 for z ∈ U and all nonzero X ∈ Cn,
r
(E) k∇r(z)k = 1,z ∈ ∂D,
(F) for every z ∈ D∩U there is a unique π(z) ∈ ∂D with
dist(z,∂D) = kz−π(z)k,
cf. [8],[9]. Note that for a function r as above and a point ζ ∈ ∂G, Taylor
expansion of r at ζ has the following form:
(2.1) r(z) = r(ζ)−2ReP(z;ζ)+L (ζ;z−ζ)+o(kz −ζk2),
r
where
n ∂r 1 n ∂2r
P(z;ζ) := − (ζ)(z −ζ )− (ζ)(z −ζ )(z −ζ )
X ∂z j j 2 X ∂z ∂z i i j j
j=1 j i,j=1 i j
is the Levi polynomial of r at ζ.
3. Proof of Theorem 1.3
We divide the proof into two parts. First we give the construction of G
t
and h (·;ζ),t ∈ T, and define the constants ε,η ,d , and d , all independecnt
t 2 1 2
of t. This is refinement of the construction from the proof of Theorem 19.1.2
from[7]. Notethatinordertogettheindependenceofalltheconstants from
t, we must be more careful here. In the second part we prove the continuity
property.
Construction of G and h (·;ζ) and the choice of ε,η ,d , and d . Fort ∈ T
t t 2 1 2
and ζ ∈ ∂G let Pc(z;ζ) be the Levi polynomial of r at ζ.
t t t
4 ARKADIUSZLEWANDOWSKI
Fix an ε > 0 such that U′ := B(ζ,ε ) ⊂⊂ U.
1 [ 1
t∈T,ζ∈∂Gt
There exists a constant C = C (t) < 1 such that
1 1
L (z;X) ≥ C kXk2, z ∈ U′,X ∈ Cn.
rt 1
Indeed, L is continuous and positive on U × (Cn \{0}), so it attains its
rt
minimum C (t) > 0 on U′ ×Sn−1. Since for any nonzero X ∈ Cn we have
1
X ∈ Sn−1, we get the required inequality. Moreover, from the assumption
kXk
(3) it follows that for s from some neighborhood of t we have
C (t)
L (z;X) ≥ 1 kXk2, z ∈U′,X ∈ Cn.
rs 2
The compactness argument then gives that C may be chosen independently
1
of t.
Taylor formula (2.1) yields that with some 0 < C < C there is
2 1
(3.1) r (z) ≥ −2ReP (z;ζ)+C kz−ζk2
t t 2
for kz −ζk < ε (t) < ε ,ζ ∈ ∂G , where ε (t) is independent of ζ ∈ ∂G
2 1 t 2 t
(and even of ζ ∈ W ⊂⊂U, some neighborhood of ∂G - see [11], Proposition
t
II.2.16). Moreover, from the proof of Theorem V.3.6 from [11] it follows that
for s close enough to t we have
C
r (z) ≥ r (ζ)−2ReP (z;ζ)+ 2kz−ζk2, ζ ∈W,kz −ζk< ε (t).
s s s 2
2
Therefore, fors nearto t, andforξ ∈ ∂G , the following estimate holds true:
s
C
r (z) ≥ −2ReP (z;ξ)+ 2kz−ξk2, kz−ξk < ε (t).
s s 2
2
The compactness argument then implies that C and ε in (3.1) may be
2 2
chosen independently of t.
Let 0 < η < ε and χ ∈ C∞(R,[0,1]) be such that χ(t) = 1 for t ≤ η1
1 2 2
and χ(t) = 0 for t ≥ η . Put χ(z;ζ) := χ(kz−ζk). This is a smooth function
1 b b
on Cbn×Cn, taking its values in [0,1]. b
Define
ϕ (z;ζ) := χ(z;ζ)P (z;ζ)+(1−χ(z;ζ))kz −ζk2, z ∈ Cn.
t t
Observe that if kz−ζk≤ η1, then ϕ (z;ζ) = P (z;ζ). Inparticular ϕ (·;ζ) ∈
2 t t t
O(B(ζ,η1)). Furthermore, for z satisfying kz −ζk ≥ η1 and r (z) < C η12
2 2 t 2 8
the following estimate holds true:
η2
(3.2) 2Reϕ (z;ζ) ≥ C 1 > 0.
t 2
8
η2
Take 0 < η < C 1 such that the connectend component G containing G
t 2 8 t t
of the open set f
G ∪{z ∈U′ :r (z) < η }
t t t
FAMILIES OF STRICTLY PSEUDOCONVEX DOMAINS AND PEAK FUNCTIONS 5
is a strictly pseudoconvex domain, relatively compact in G ∪U′. Because of
t
the assumption (3), there exists a positive number β such that for s close to
t the connected component G containing G of the set
s s
›
G ∪{z ∈ U′ : r (z) < η −β}
s s t
is a strictly pseudoconvex domain, relatively compact in G ∪ U′. Making
s
again use of the compactness of T, we conclude that in fact η = η may be
t
taken independently of t. Note that for the family (G ) the assumption
t t∈T
(3) remains true. f
The function ϕ (·;ζ) ∈ C∞(Cn) does not vanish on G\B(ζ,η1) and is in
t 2
O(B(ζ,η1)). Therefore ∂¯ 1 defines a ∂¯-closed C∞ fo‹rm
2 ϕt(·;ζ)
n
α (·;ζ) = α (·;ζ)dz
t X t,j j
j=1
on G , where
t
f
0, z ∈ G ∩B(ζ;η1),
α =  t 2
t,j −∂ϕt(z;ζ)· 1 , z ∈ Gf \B(ζ;η1).
 ∂z¯j ϕ2t(z;ζ) ft 2
Thanksto(3.2)wehavekα (·;ζ)k ≤ C ,where,utilizingthecompactness
t,j Gt 3
ofT together withthe assumption (‹3), we deliver that C3 is independent of t
and ζ ∈∂G . [11, Theorem V.2.7] gives then the functions v (·;ζ) ∈ C∞(G )
t t t
with ∂¯v (·;ζ) = α (·;ζ) and f
t t
kv (·;ζ)k ≤ C ,
t Gt 4
‹
where C does not depend on ζ ∈ ∂G . Moreover, by [11, Theorem V.3.6]
4 t
and the compactness of T, C may be chosen to be independent of t.
4
Define
1
f (·;ζ) := (·;ζ)+C −v (·;ζ), z ∈ G \Z (ζ),
t 4 t t t
ϕ
t f
where
Z (ζ):= {z ∈G :ϕ (z;ζ) = 0}.
t t t
f
Then f (·;ζ) ∈ O(G \Z (ζ)) as well as
t t t
f
Ref (·;ζ) > 0
t
on the set (G \B(ζ,η1))∪(G \{ζ}), in virtue of (3.1) and (3.2). Since for
t 2 t
any ζ 6=z ∈f∂G ∩B(ζ,η1) there exists a neighborhood U of z such that
0 t 2 z0 0
Ref (·;ζ) > 0 on U , we conclude that there exists a neighborhhod U of
t z0 t,ζ
G \{ζ} such that the function
t
h (·;ζ) := exp(−g (·;ζ)),
t t
where g (·;ζ) := 1 , is holomorphic on H := (G \B(ζ,η1))∪U . Note
t ft(·;ζ) t,ζ t 2 t,ζ
that h takes its values in D. f
t
6 ARKADIUSZLEWANDOWSKI
There exists a C > 0, independent on t, such that
5
|P (z;ζ)| ≤ C kz−ζk, ζ,z ∈ U′.
t 5
Therefore, since for 0 < η2 < min¶η21,4C41C5©, which now is independent of
t, and forz ∈ ((H ∪B(ζ,η ))∩B(ζ,η ))\Z (ζ)the following equality holds
t,ζ 2 2 t
true:
P (z;ζ)
t
g (z;ζ) = ,
t
1−P (z;ζ)(v (z;ζ)−C )
t t 4
we conclude that g (·;ζ) is bounded near Z (ζ), which yields it extends to
t t
be holomorphic on H := H ∪B(ζ;η ).
t,ζ t,ζ 2
fi
Now H depends on ζ, but using the inclusion G ⊂ H , we may find
t,ζ t t,ζ
fi fi
some G , strictly pseudoconvex domain which is independent on ζ ∈ ∂G ,
t t
such tchat G ⊂ G ⊂ H for each ζ ∈ ∂G , and with the property that
t t t,ζ t
fi
h (·;ζ) ∈ O(G ),cζ ∈ ∂G (use the joint continuity of ϕ with respect to z
t t t t
and ζ to shricnk H little bit to get some domain with desired properties,
t,ζ
fi
independent on ξ close to ζ, and finally apply the compactness of ∂G ).
t
LetC ,independentontandζ ∈ ∂G ,suchthatforz ∈ G withkz−ζk < η
6 t t 2
we have c
C kz−ζk
5
g (z;ζ) ≤ ≤ C kz−ζk.
t 6
1−2C C kz−ζk
4 5
This implies
|1−h (z;ζ)| ≤ C |g (z;ζ)| ≤ C C kz−ζk =:d kz−ζk
t 7 t 6 7 1
for z ∈ G ,kz −ζk < η ,ζ ∈ ∂G , if only C is chosen so that
t 2 t 7
c
|eλ−1| ≤ C |λ|, |λ| ≤ C η .
7 6 2
In particular, d does not depend on t and we have h (ζ;ζ)= 1.
1 t
Furthermore, for z ∈G ,kz−ζk≥ η there is
t 1
1+kz−ζk2(C −Rev (z;ζ))
Reg (z;ζ) = kz−ζk2 4 t
t |1−kz−ζk2(v (z;ζ)−C )|2
t 4
η2
≥ 1 =: C ,
(1+2(diamU)2C )2 8
4
which gives
(3.3) |h (z;ζ)| ≤ e−C8 =:d < 1.
t 2
Observe that d is independent on t. (cid:3)
2
Proof of continuity. Fix α > 0,t ∈ T,ζ ∈ ∂G , and z ∈ G . Let K be
0 0 t0 0 t0 0
a compact subset of G , containing in its interior the set Gd∪{z }. In the
t0 t0 0
sequelwe shall use thge following convention: whenever we say that the triple
(s,ξ,w) is near to (t ,ζ ,z ), it will carry the additional information that
0 0 0
ξ ∈ ∂G ,w ∈ G , unless explicitly stated otherwise.
s s
”
FAMILIES OF STRICTLY PSEUDOCONVEX DOMAINS AND PEAK FUNCTIONS 7
Observethatfor(s,ξ)closeto(t ,ζ )(evenwithoutrequiringthatξ ∈ ∂G ),
0 0 s
and any z,w ∈U′ we have
|P (z;ζ )−P (w;ξ)| < M α
t0 0 s 1
with some positive M . In particular, the same estimate is true for z = z
1 0
and w close to z .
0
Further, using the fact that all the functions ϕ are continuous as functions
t
of both variables, we conclude that for (s,ξ) close to (t ,ζ ) we have
0 0
kϕt0(·;ζ0)−ϕs(·;ξ)kU′ < M2α
with some positive M .
2
For (s,ξ) near (t ,ζ ) we have
0 0
∂ϕ ∂ϕ
t0(·;ζ )− s(·;ξ) < M α
(cid:13)(cid:13) ∂z¯j 0 ∂z¯j (cid:13)(cid:13)U′ 3
(cid:13) (cid:13)
with some positive M . Furthermore, for z ∈ G ∩G the following estimates
3 s t
›
hold true: f
(I) If z ∈/ B(ζ , η1)∪B(ξ,η1), then
0 2 2
|α (z;ζ )−α (z;ξ)| < Lα,
t0,j 0 s,j
where positive constant L does not depend on z as above. Indeed,
∂ϕt0(z;ζ ) ∂ϕs(z;ξ) ϕ2(z;ξ)∂ϕt0(z;ζ )−ϕ2 (z;ζ )∂ϕs(z;ξ)
(cid:12)(cid:12) ∂z¯j 0 − ∂z¯j (cid:12)(cid:12) = (cid:12)(cid:12) s ∂z¯j 0 t0 0 ∂z¯j (cid:12)(cid:12)
(cid:12) ϕ2 (z;ζ ) ϕ2(z;ξ) (cid:12) (cid:12) ϕ2 (z;ζ )ϕ2(z;ξ) (cid:12)
(cid:12) t0 0 s (cid:12) (cid:12) t0 0 s (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) 64 (cid:12) (cid:12)∂ϕ ∂ϕ (cid:12)
≤ ϕ2(z;ξ) t0(z;ζ )−ϕ2 (z;ζ ) s(z;ξ)
C2η4(cid:12) s ∂z¯ 0 t0 0 ∂z¯ (cid:12)
2 1(cid:12) j j (cid:12)
(cid:12) (cid:12)
64 ∂ϕ ∂ϕ ∂ϕ
≤ C22η14kϕ2skU′(cid:13)(cid:13) ∂z¯tj0(·;ζ0)− ∂z¯js(·;ξ)(cid:13)(cid:13)U′ +(cid:13)(cid:13)∂z¯js(·;ξ)(cid:13)(cid:13)U′kϕ2s −ϕ2t0k
(cid:13) (cid:13) (cid:13) (cid:13)
64
≤ L M α+L M α =:Lα,
C2η4 1 3 2 2
2 1
where the first inequality is the consequence of (3.2).
(II) If z ∈ B(ζ , η1)∪B(ξ,η1):
0 2 2
Observe that letting ξ close to ζ , we may make the balls arbitrarily close
0
each other. Using then the assumption (3), the fact that η were chosen tobe
η2
strictly smaller than C 1, and the strictness of uniform estimate (3.2), we
2 8
seethat for (s,ξ) close enough to (t ,ζ )the estimate similar to the previous
0 0
one holds true for z ∈ S := B(w,γ) with some sufficiently small
[
w:kw−ζ0k=η21
γ > 0, (and is independent on such z). Additionally, (s,ξ) may be chosen so
that S′ := (B(ζ ,η1)∪B(ξ, η1))\S ⊂ B(ζ ,η1)∩B(ξ, η1).
0 2 2 0 2 2
8 ARKADIUSZLEWANDOWSKI
Noting that for z ∈ S′ and (s,ξ) as above α (z;ζ ) = α (z;ξ) = 0, we
t0,j 0 s,j
conclude that
kα (·;ζ )−α (·;ξ)k ≤ M α
t0 0 s Gt0∩Gs 4
with some positive M . › f
4
Ofcourse G ⊂ G . This yields that for s close to t we have G ⊂ G as
t0 t0 0 t0 s
›
well as G ⊂ G (gthe assumption (3) remains true for the family (G ) ).
s t0 t t∈T
ForsclosetotgwemaynowpicksomeG ,astrictlypseudoconvexfdomain
0 t0,s
with smooth boundary and such that
G ∪G ⊂ K ⊂⊂G ⊂⊂G ∩G .
s t0 0 t0,s s t0
›
g
Again thanks to the property (3), G may be chosen independently of s if
t0,s
s is close enough to t . For such s, denote it by Gt0. Then, using Lemma 2
0
from [6], we find some positive constant Γ such that
kv (·;ζ )−v (·;ξ)k ≤ Γkα (·;ζ )−α (·;ξ)k ≤ ΓM α =:M α.
t0 0 s K0 t0 0 s Gt0 4 5
Consequently, for (s,ξ,w) close to (t ,ζ ,z ) there is
0 0 0
|v (z ;ζ )−v (w;ξ)| ≤ |v (z ;ζ )−v (w;ζ )|+|v (w;ζ )−v (w;ξ)| ≤ M α
t0 0 0 s t0 0 0 t0 0 t0 0 s 6
for some positive M (use the smoothness of v (·;ζ )).
6 t0 0
There are two cases to be considered:
Case 1. z ∈ H ∩intK .
0 t0,ζ0 0
Then ϕ (z ;ζ ) 6= 0 and for (s,ξ,w) near (t ,ζ ,z ) we have ϕ (w;ξ) 6= 0.
t0 0 0 0 0 0 s
For such (s,ξ,w) we have
1 1
|f (z ;ζ )−f (w;ξ)| ≤ − +|v (z ;ζ )−v (w;ξ)|
t0 0 0 s (cid:12)ϕ (z ;ζ ) ϕ (w;ξ)(cid:12) t0 0 0 s
(cid:12) t0 0 0 s (cid:12)
(cid:12) (cid:12)
ϕ (w;ξ)−ϕ (z ;ζ )
≤ s t0 0 0 +M α.
(cid:12) ϕ (z ;ζ )ϕ (w;ξ) (cid:12) 6
(cid:12) t0 0 0 s (cid:12)
(cid:12) (cid:12)
Considering the last but one term, its denominator is bounded below by
some positive constant for (s,ξ,w) close to (t ,ζ ,z ), and the counter is
0 0 0
estimated from above by M α. Thus for (s,ξ,w) close to (t ,ζ ,z )
2 0 0 0
|f (z ;ζ )−f (w;ξ)| ≤ M α
t0 0 0 s 7
for some positive M .
7
Inour situation the function g (·;ζ ) is holomorphic in aneighborhood of
t0 0
z and so is g (·;ξ) for (s,ξ) close to (t ,ζ ). We conclude that for (s,ξ,w)
0 s 0 0
close to (t ,ζ ,z ) there is
0 0 0
|g (z ;ζ )−g (w;ξ)| ≤ M α
t0 0 0 s 8
for some positive M , and
8
|h (z ;ζ )−h (w;ξ)| = exp(−g (z ;ζ ))−exp(−g (w;ξ)) ≤ M α
t0 0 0 s (cid:12) t0 0 0 s (cid:12) 9
(cid:12) (cid:12)
(cid:12) (cid:12)
FAMILIES OF STRICTLY PSEUDOCONVEX DOMAINS AND PEAK FUNCTIONS 9
for some positive M .
9
Case 2. z ∈ (G ∩B(ζ ,η ))∩intK .
0 t0 0 2 t0
g
(I) Suppose ϕ (z ;ζ ) 6= 0.
t0 0 0
It is equivalent to P (z ;ζ ) 6= 0. This yields that P (w;ξ) 6= 0 for (s,ξ,w)
t0 0 0 s
close to (t ,ζ ,z ). Then
0 0 0
|g (z ;ζ )−g (w;ξ)|
t0 0 0 s
P (z ;ζ ) P (w;ξ)
= t0 0 0 − s
(cid:12)1−P (z ;ζ )(v (z ;ζ )−C ) 1−P (w;ξ)(v (w;ξ)−C )(cid:12)
(cid:12) t0 0 0 t0 0 0 4 s s 4 (cid:12)
(cid:12) (cid:12)
≤N|P (z ;ζ )−P (w;ξ)|+N|P (z ;ζ )P (w;ξ)kv (z ;ζ )−v (w;ξ)|
t0 0 0 s t0 0 0 s t0 0 0 s
≤ NM α+NM α =:M α,
1 6 10
and similarly as in the previous case
|h (z ;ζ )−h (w;ξ)| ≤ M α
t0 0 0 s 11
with some positive N,M , and M .
10 11
(II) Suppose ϕ (z ;ζ ) = 0.
t0 0 0
This is equivalent to P (z ;ζ ) = 0. Then for some positive ρ we have
t0 0 0
B(z ,ρ) ⊂⊂ K ∩ B(ζ ,η ). Similarly, for (s,ξ) close to (t ,ζ ) there is
0 0 0 2 0 0
B(z ,ρ) ⊂⊂ K ∩B(ξ,η ). Therefore, because of the choice of d in (3.3), for
0 0 2 2
(s,ξ,w) close to (t ,ζ ,z ),w ∈ B(z ,ρ/2) there is
0 0 0 0
|h (w;ζ )−h (w;ξ)| ≤ |1−h (w;ζ )|+|1−h (w;ξ)|
t0 0 s t0 0 s
≤ d (kw−ζ k+kw−ξk) ≤ 2d η .
1 0 1 2
Consequently, since the functions h (·;ζ )−h (·;ξ) are holomorphic in suit-
t0 0 s
able neighborhood of z for (s,ξ) close to (t ,ζ ), for some positive ρ < ρ/2,
0 0 0
for every x,y ∈ B(z0,ρ/2) we have e
e
(3.4) |h (x;ζ )−h (x;ξ)−h (y;ζ )+h (y;ξ)| ≤ α.
t0 0 s t0 0 s
Moreover, ρ may be chosen so that for v,w ∈ B(z ,ρ/2) there is
0
e e
(3.5) |h (v;ζ )−h (w;ζ )| ≤ α,
t0 0 t0 0
by continuity of h (·;ζ ).
t0 0
Fix some w ∈ B(z ,ρ/2) such that P (w ;ζ ) 6= 0. Then for (s,ξ) near
0 0 t0 0 0
(t0,ζ0), by virtue of thee subcase (I), we have
|h (w ;ζ )−h (w ;ξ)| ≤ α.
t0 0 0 s 0
Finally, for w ∈ B(z ,ρ/2) and (s,ξ) close to (t ,ζ ) we have
0 0 0
e
|h (z ;ζ )−h (w;ξ)| ≤ |h (z ;ζ )−h (w ;ζ )|+|h (w ;ζ )−h (w ;ξ)|
t0 0 0 s t0 0 0 t0 0 0 t0 0 0 s 0
+|h (w ;ξ)−h (w;ξ)| ≤ α+α+2α = 4α,
s 0 s
10 ARKADIUSZLEWANDOWSKI
where the last estimate follows from (3.4) and (3.5), which leads us to the
conclusion. (cid:3)
References
[1] E. Bedford, J.E. Fornæss, Biholomorphic maps of weakly pseudoconvex domains,
DukeMath. J. 45 (1978), 711-719.
[2] F. Deng, Q. Guan, L. Zhang, Properties of squeezing functions and global transfor-
mations of bounded domains, Trans. Amer.Math. Soc. 368 (4) (2016), 2679-2696.
[3] B. DrinovecDrnovšek,Complete proper holomorphic embeddings of strictly pseudo-
convex domains into balls, J. Math. Anal. Appl.431 (2015) 2, 705-713.
[4] J.E. Fornæss, Sup-norm estimates for ∂ in C2, Ann.of Math. 123 (1986), 335-345.
[5] F. Forstnerič, Embedding strictly pseudoconvex domains into balls, Trans. Amer.
Math. Soc. 295 (1986), 347-368.
[6] I. Graham, Boundary behavior of the Carathéodory and Kobayashi metrics on
strongly pseudoconvex domains in Cn with smooth boundary, Trans. Amer. Math.
Soc. 207 (1975), 219-240.
[7] M. Jarnicki, P. Pflug, Invariant Distances and Metrics in Complex Analysis, 2nd
edition, de GruyterExpositions in Mathematics 9, Walter deGruyter2014.
[8] S.G. Krantz, Function Theory of Several Complex, reprint of the 1992 ed, AMS
Chelsea Publishing, Providence, RI,2001.
[9] S.G. Krantz, H.R. Parks, Distance to Ck hypersurfaces, Journal of Diff. Equ. 40
(1981), 116-120.
[10] R.M. Range, The Carathéodory metric and holomorphic maps on a class of weakly
pseudoconvex domains, Pacific J. Math. 78 (1978), 173-188.
[11] R.M. Range, Holomorphic Functions and Integral Representations in Several Com-
plex Variables, GraduateTexts in Mathematics, 108, SpringerVerlag, 1986.
Instituteof Mathematics, Facultyof Mathematics and Computer Science,
Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
E-mail address: [email protected]