Discrete sequences in unbounded domains

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by  Alberto Saracco

DISCRETE SEQUENCES IN UNBOUNDED DOMAINS
ALBERTO SARACCO
Abstract. Discrete sequences with respect to the Kobayashi distance
in a strongly pseudoconvex bounded domain D are related to Carleson
6 measuresbyaformulathatusestheEuclideandistancefromthebound-
1 ary of D.
0
Thusthespeedofescapeattheboundaryofsuchsequencehasbeen
2 studied in details for strongly pseudoconvex boundeddomain D.

n In this note we show that such estimations completely fail if the
a domain is not bounded.
J

5
1
Let (X,d) be a metric space. A sequence Γ = {x } ⊂ X of points in X is
j
] uniformly discrete if there exists δ > 0 such that d(xj,xk) ≥ δ for all j 6= k.
V
In this case inf d(x ,x ) is the separation constant of Γ.
j k
C j6=k
h. Finite unions of uniformly discrete sequences in a strongly pseudoconvex
t boundeddomainD ⊂ Cn withrespecttotheKobayashidistanceareamean
a
m to produce explicit examples of discrete Carleson measures.
[ Theorem0.1. LetD ⋐ Cn beastronglypseudoconvex boundeddomain, and

1 let Γ = {zj}j∈N be a sequence in D. Then Γ is a finite union of uniformly
v
discrete (with respect to the Kobayashi distance) sequences if and only if
6
3
8 d(zj,∂D)n+1δzj
3 zXj∈Γ
0
. is a Carleson measure of Ap(D), where δ is the Dirac measure at z and
1 zj j
0 d(·,∂D) is the Euclidean distance from the boundary.
6
1 Thistheorem,byAbateandtheauthor[1]isageneralization ofresultsby
v: Massaneda [7], Jevti´c, Massaneda and Thomas [5] and Duren and Weir [4].
i These results motivated the study of the boundary behaviour, with respect
X
to the Euclidean distance, of uniformly discrete sequences (with respect
r
a to the Kobayashi distance) in a strongly pseudoconvex bounded domain
D ⊂ Cn. In [3], [4], [6] and [1] were obtained results in this direction for D
being the unit ball or a strongly pseudoconvex bounded domain or even a
hyperbolic domain with finite Euclidean volume.
WeciteheretwotheoremsprovedbyAbateandtheauthorin[1]. Thefirst
one generalizes the results obtained by Duren, MacCluer, Schuster, Vukoti´c
and Weir for the unitball, to strongly pseudoconvex boundeddomains (and
is a sharp estimation). The second one is valid in a more general setting but
Date: January 18, 2016.
2000 Mathematics Subject Classification. Primary 32Q45, Secondary 32T15, 52A20.
Key words and phrases. uniformly discrete sequences · unboundeddomains.
1
2 ALBERTOSARACCO
provides a worst estimation of the boundarybehaviour of uniformlydiscrete
sequences.
Theorem 0.2. Let D ⋐ Cn be a strongly pseudoconvex bounded domain.
Let Γ = {z } ⊂ D be an uniformly discrete sequence (with respect to the
j
Kobayashi distance) with d(z ,∂D) < 1 for all z ∈ Γ. Then
j j
1
d(z ,∂D)nh − < +∞
j
(cid:18) logd(z ,∂D)(cid:19)
zXj∈Γ j
for any increasing function h: R+ → R+ such that
+∞
1
h < +∞ .
(cid:18)m(cid:19)
mX=1
Theorem 0.3. Let D ⊂ Cn be a hyperbolic domain with finite Euclidean
volume.
Let Γ = {z } ⊂ D be an uniformly discrete sequence (with respect to the
j
Kobayashi distance) with d(z ,∂D) < 1 for all z ∈ Γ. Then
j j
1
d(z ,∂D)2nh − < +∞
j
(cid:18) logd(z ,∂D)(cid:19)
zXj∈Γ j
for any increasing function h: R+ → R+ such that
+∞
1
h < +∞ .
(cid:18)m(cid:19)
mX=1
One might hope that the boundedness (or limited volume) condition is
unnecessary, at least in cases where the hyperbolicity of the domain imply
some form of boundedness. For example, (complete) hyperbolic convex (or
even C-convex) domains are known to be biholomorphically equivalent to a
bounded domain (see [2] for the convex case, [8] and [9] for the C-convex
case).
If such results were true this would set a much general result for any con-
vex (respectively C-convex) domain, since such a domain D ⊂ Cn is linearly
equivalent to the product of Cn−k and a (complete) hyperbolic convex (re-
spectively C-convex) domain D′ ⊂ Ck and the Kobayashi pseudodistance in
D would coincide with the Kobayashi distance on the hyperbolic factor D′
(see again [2], [8] and [9]).
Unfortunately, this is actually not possible. We give an example showing
that for the half plane H ⊂ C and then generalize the result to a huge class
of unbounded domains.
1. An example
Let D ⊂ C be the unit disc. D is biholomorphic to the right half plane
H = {z ∈ C | Rez > 0} via the function
1−z
ϕ(z) = .
1+z
DISCRETE SEQUENCES IN UNBOUNDED DOMAINS 3
Let us consider the set of points at a fixed Euclidean distance ε> 0 from
the boundary of H:
l = {ε+iy | y ∈ R}.
δ
Its image in ∆ via ϕ−1 is an ellipse E internally tangent to ∂∆ of center
ε
ε
C = − .
1+ε
Let us fix a δ > 0. We define a discrete sequence of points Γ = {zj}j∈N in ∆
as follows. Fix z ∈∆∩E . Since E is tangent to ∂∆, for every k ∈ N the
0 ε ε
set E = E ∩{z ∈ ∆ | d(z ,z) = kδ} is not empty. Hence we can chose
k,ε ε 0
z ∈ E and by the triangular inequality d(z ,z ) ≥ δ for every h 6= k ∈ N,
k k,ε h k
i.e. Γ = {zj}j∈N is a uniformly discrete sequence with separation constant
δ. The points z approach the boundary of ∆, as j goes to infinite.
j
If we look at the very same sequence in H, i.e. if we define w = ϕ−1(z ),
j j
then ϕ−1(Γ) = {wj}j∈N is a uniformly discrete sequence in H with separa-
tion constant δ, since ϕ : H → ∆ is a biholomorphism, and d(w ,∂H) = ε
j
for each j ∈ N.
Hence if
F: R+ → R+
is any function such that F(x) > 0 for every x > 0, then
∞
F(d(w ,∂H)) = F(ε) = +∞.
j
wj∈Xϕ−1(Γ) Xj=0
In particular a result such as Theorem 0.2 or 0.3, valid in D, fails in H,
despite the two domains being biholomorphic. This is obviously because
Euclidean distance from boundary is not preserved by biholorrphisms.
2. A theorem
The argument of the previous example can be generalized to prove the
following
Theorem 2.1. Let D ⊂ Cn be a Kobayashi hyperbolic domain such that
there exists c > 0 for which D = {z ∈ D | d(z,∂D) ≥ c} ⊂ D is
c
non compact. Then for any δ > 0 there exists a uniformly discrete sequence
Γ = {zj}j∈N in D with separation constant δ such that Γ ⊂ Dc. In particular
F(d(w ,∂D)) = +∞,
j
wj∈Xϕ−1(Γ)
for every increasing function F: R+ → R+ such that F(x) > 0 for every
x > 0.
Remark 2.1. Thus no equivalent of theorems 0.2 or 0.3 holds for such a
domain.
Remark 2.2. NotethatifthesetD isnoncompact, i.e.unboundedsinceit
c
isclosed,thenD\D hasnotafiniteEuclideanvolume,hencethehypothesis
c
of theorem 0.3 does not hold.
4 ALBERTOSARACCO
Proof. Let c > 0 be as in the hypothesis. Fix δ > 0. We will construct
the required uniformly discrete sequence Γ (with separation constant δ) by
induction.
Since D is non compact, it is not empty. Fix z ∈D .
c 0 c
Let k ∈ N. Suppose for all j < k the points z ∈ D has been choosen
j c
in such a way that any two such points are at least δ-apart from each other
with respect to the Kobayashi distance. Since D is Kobayashi hyperbolic,
for every j < k B = B (z ), the Kobayashi ball of radius δ and center z
j δ j j
is relatively compact in D. Hence ∪ B is relatively compact in D and thus
j j
D \∪ B is not empty. By choosing z ∈ D \∪ B , z is at least δ-apart
c j j k c j j k
from each other z , j < k, and the inductive step is done.
j
Hence the claimed Γ is constructed. (cid:3)
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Dipartimento di Matematica e Informatica, Universita` di Parma, Parco
Area delle Scienze 53/A, I-43124 Parma, Italy
E-mail address: [email protected]