Discrete sequences with respect to the Kobayashi distance in a strongly
pseudoconvex bounded domain $D$ are related to Carleson measures by a formula
that uses the Euclidean distance from the boundary of $D$.
  Thus the speed of escape at the boundary of such sequence has been studied in
details for strongly pseudoconvex bounded domain $D$.
  In this note we show that such estimations completely fail if the domain is
not bounded.Comment: 4 page

Saracco, Alberto

English

arXiv

Discrete sequences with respect to the Kobayashi distance
in a strongly pseudoconvex bounded domain D are related to Carleson
measures by a formula that uses the Euclidean distance from the boundary
of D.
Thus the speed of escape at the boundary of such sequence has been
studied in details for strongly pseudoconvex bounded domain D.
In this note we show that such estimations completely fail if the
domain is not bounded

SARACCO, Alberto

Archivio istituzionale della Ricerca -  Università degli Studi di Parma

12 October 2022University of Parma Research RepositoryDISCRETE SEQUENCES IN UNBOUNDED DOMAINS / Saracco, Alberto. - In: SIBIRSKIE ÈLEKTRONNYEMATEMATIčESKIE IZVESTIÂ.. - ISSN 1813-3304. - 14:(2017), pp. 22-25. [10.17377/semi.2017.14.03]OriginalDISCRETE SEQUENCES IN UNBOUNDED DOMAINSPublisher:PublishedDOI:10.17377/semi.2017.14.03Terms of use:openAccessPublisher copyright(Article begins on next page)Anyone can freely access the full text of works made available as "Open Access". Works made availableAvailability:This version is available at: 11381/2825592 since: 2018-02-12T09:37:25ZThis is the peer reviewd version of the followng article:S e⃝MR ISSN 1813-3304СИБИРСКИЕ ЭЛЕКТРОННЫЕМАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯSiberian Electronic Mathematical Reportshttp://semr.math.nsc.ruТом 14, стр. 22–25 (2017) УДК 514.7DOI 10.17377/semi.2017.14.03 MSC 32Q45, 32T15, 52A20DISCRETE SEQUENCES IN UNBOUNDED DOMAINSA.SARACCOAbstract. Discrete sequences with respect to the Kobayashi distancein a strongly pseudoconvex bounded domain D are related to Carlesonmeasures by a formula that uses the Euclidean distance from the boundaryof D.Thus the speed of escape at the boundary of such sequence has beenstudied in details for strongly pseudoconvex bounded domain D.In this note we show that such estimations completely fail if thedomain is not boundedKeywords: uniformly discrete sequences, unbounded domains.Let (X, d) be a metric space. A sequence Γ = {xj} ⊂ X of points in X isuniformly discrete if there exists δ > 0 such that d(xj , xk) ≥ δ for all j ̸= k.In this case infj ̸=kd(xj , xk) is the separation constant of Γ.Finite unions of uniformly discrete sequences in a strongly pseudoconvex boundeddomain D ⊂ Cn with respect to the Kobayashi distance are a mean to produceexplicit examples of discrete Carleson measures.Theorem 1. Let D b Cn be a strongly pseudoconvex bounded domain, and letΓ = {zj}j∈N be a sequence in D. Then Γ is a finite union of uniformly discrete(with respect to the Kobayashi distance) sequences if and only if∑zj∈Γd(zj , ∂D)n+1δzjis a Carleson measure of Ap(D), where δzj is the Dirac measure at zj and d(·, ∂D)is the Euclidean distance from the boundary.Saracco, A., Discrete sequences in unbounded domains.c⃝ 2017 Saracco A.Received September, 4, 2014, published January, 16, 2017.22DISCRETE SEQUENCES IN UNBOUNDED DOMAINS 23This theorem, by Abate and the author [1] is a generalization of results byMassaneda [7], Jevtić, Massaneda and Thomas [5] and Duren and Weir [4]. Theseresults motivated the study of the boundary behaviour, with respect to the Euclideandistance, of uniformly discrete sequences (with respect 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 stronglypseudoconvex bounded domain or even a hyperbolic domain with finite Euclideanvolume.We cite here two theorems proved by Abate and the author in [1]. The firstone generalizes the results obtained by Duren, MacCluer, Schuster, Vukotić andWeir for the unit ball, to strongly pseudoconvex bounded domains (and is a sharpestimation). The second one is valid in a more general setting but provides a worstestimation of the boundary behaviour of uniformly discrete sequences.Theorem 2. Let D b Cn be a strongly pseudoconvex bounded domain.Let Γ = {zj} ⊂ D be an uniformly discrete sequence (with respect to the Kobayashidistance) with d(zj , ∂D) < 1 for all zj ∈ Γ. Then∑zj∈Γd(zj , ∂D)n h(− 1log d(zj , ∂D))< +∞for any increasing function h : R+ → R+ such that+∞∑m=1h(1m)< +∞ .Theorem 3. Let D ⊂ Cn be a hyperbolic domain with finite Euclidean volume.Let Γ = {zj} ⊂ D be an uniformly discrete sequence (with respect to the Kobayashidistance) with d(zj , ∂D) < 1 for all zj ∈ Γ. Then∑zj∈Γd(zj , ∂D)2n h(− 1log d(zj , ∂D))< +∞for any increasing function h : R+ → R+ such that+∞∑m=1h(1m)< +∞ .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 knownto be biholomorphically equivalent to a bounded domain (see [2] for the convexcase, [8] and [9] for the C-convex case).If such results were true this would set a much general result for any convex(respectively C-convex) domain, since such a domain D ⊂ Cn is linearly equivalentto the product of Cn−k and a (complete) hyperbolic convex (respectively C-convex)domain D′ ⊂ Ck and the Kobayashi pseudodistance in D would coincide with theKobayashi distance on the hyperbolic factor D′ (see again [2], [8] and [9]).Unfortunately, this is actually not possible. We give an example showing that forthe half plane H ⊂ C and then generalize the result to a huge class of unboundeddomains.24 A.SARACCO1. An exampleLet D ⊂ C be the unit disc, d be the Euclidean distance in the plane and kD theKobayashi (i.e. Poincaré) distance in the disc. D is biholomorphic to the right halfplane H = {z ∈ C | Re z > 0} via the functionφ(z) =1− z1 + z.Let us consider the set of points at a fixed Euclidean distance ε > 0 from theboundary of H:lδ = {ε+ iy | y ∈ R} .Its image in D via φ−1 is a circle Eε internally tangent to ∂D of centerC = − ε1 + ε.Let us fix a δ > 0. We define a discrete sequence of points Γ = {zj}j∈N in Das follows. Fix z0 ∈ D ∩ Eε. Since Eε is tangent to ∂D, for every k ∈ N the setEk,ε = Eε∩{z ∈ D | kD(z0, z) = kδ} is not empty. Hence we can chose zk ∈ Ek,ε andby the triangular inequality kD(zh, zk) ≥ δ for every h ̸= k ∈ N, i.e. Γ = {zj}j∈N isa uniformly discrete sequence with separation constant δ. The points zj approachthe boundary of D, as j goes to infinite.If we look at the very same sequence in H, i.e. if we define wj = φ−1(zj), thenφ−1(Γ) = {wj}j∈N is a uniformly discrete sequence in H with separation constantδ, since φ : H → D is a biholomorphism, and d(wj , ∂H) = ε for each j ∈ N.Hence ifF : R+ → R+is any function such that F (x) > 0 for every x > 0, then∑wj∈φ−1(Γ)F (d(wj , ∂H)) =∞∑j=0F (ε) = +∞ .In particular a result such as Theorem 2 or 3, valid in D, fails in H, despite thetwo domains being biholomorphic. This is obviously because Euclidean distancefrom boundary is not preserved by biholorrphisms.2. A theoremThe argument of the previous example can be generalized to prove the followingTheorem 4. Let D ⊂ Cn be a Kobayashi hyperbolic domain such that there existsc > 0 for which Dc = {z ∈ D | d(z, ∂D) ≥ c} ⊂ D is non compact. Then for anyδ > 0 there exists a uniformly discrete sequence Γ = {zj}j∈N in D with separationconstant δ such that Γ ⊂ Dc. In particular∑wj∈φ−1(Γ)F (d(wj , ∂D)) = +∞ ,for every increasing function F : R+ → R+ such that F (x) > 0 for every x > 0.Remark 1. Thus no equivalent of theorems 2 or 3 holds for such a domain.Remark 2. Note that if the set Dc is non compact, i.e. unbounded since it is closed,then D \Dc has not a finite Euclidean volume, hence the hypothesis of theorem 3does not hold.DISCRETE SEQUENCES IN UNBOUNDED DOMAINS 25Proof. Let c > 0 be as in the hypothesis. Fix δ > 0. We will construct the requireduniformly discrete sequence Γ (with separation constant δ) by induction.Since Dc is non compact, it is not empty. Fix z0 ∈ Dc.Let k ∈ N. Suppose for all j < k the points zj ∈ Dc has been choosen in such away that any two such points are at least δ-apart from each other with respect to theKobayashi distance. Since D is Kobayashi hyperbolic, for every j < k Bj = Bδ(zj),the Kobayashi ball of radius δ and center zj is relatively compact in D. Hence∪jBj is relatively compact in D and thus Dc \ ∪jBj is not empty. By choosingzk ∈ Dc \ ∪jBj , zk is at least δ-apart from each other zj , j < k, and the inductivestep is done.Hence the claimed Γ is constructed. References[1] M. Abate, A. Saracco, Carleson measures and uniformly discrete sequences in stronglypseudoconvex domains, J. London Math. Soc. 84 (2011), 587–605. MR2802500[2] F. Bracci, A. Saracco, Hyperbolicity in unbounded convex domains, Forum Math., 5 (2009),815–826. MR2560392[3] P. Duren, A. Schuster, D. Vukotić, On uniformly discrete sequences in the disk, in Quadraturedomains and applications, eds. P. Ebenfeld, B. Gustafsson, D. Khavinson and M. Putinar,Birkhäuser, Basel, 2005, 131–150. MR2129739[4] P. Duren, R. Weir, The pseudohyperbolic metric and Bergman spaces in the ball, Trans. Amer.Math. Soc. 359:1 (2007), 63–76. MR2247882[5] M. Jevtić, X. Massaneda, P. J. Thomas, Interpolating sequences for weighted Bergman spacesof the ball, Michigan Math. J., 43 (1996), 495–517. MR1420589[6] B.D. MacCluer, Uniformly discrete sequences in the ball, J. Math. Anal. Appl., 318 (2006),37–42. MR2210869[7] X. Massaneda, A−p interpolation in the unit ball, J. London Math. Soc. 52 (1995), 391–401.MR1356150[8] N. Nikolov, P. Pflug, W. Zwonek, An example of a bounded C-convex domain which is notbiholomorphic to a convex domain, Math. Scand. 102:1 (2008), 149–155. MR2420684[9] N. Nikolov, A. Saracco, Hyperbolicity of unbounded C-convex domains, C. R. Acad. BulgareSci. 60:9 (2007), 737–748. MR2368753Alberto Saracco,Dipartimento di Matematica e Informatica,Università di Parma,Parco Area delle Scienze 53/A, I-43124 Parma, ItalyE-mail address: alberto.saracco@unipr.it

Discrete sequences in unbounded domains

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