Skip to main content
SpringerLink
Log in

A uniqueness theorem and the myrberg phenomenon for a Zalcman domain

  • Published:
Journal d’Analyse Mathématique Aims and scope

Abstract

LetR=Δ0\∪nΔn be a Zalcman domain (or L-domain), where Δ0 : 0<|z| <1, Δn : |z-c n|≤r n,cn ↘0, Δn ⊂ Δ0 and Δn ∩ Δm= φ(n≠m). 0217 0115 V 3 For an unlimited two-sheeted covering\(\tilde R = \varphi ^{ - 1} (R)\) with the branch points {φ-1(c n)}, set\(\tilde R = \varphi ^{ - 1} (R)\). In the casec n=2−n, it was proved that if a uniqueness theorem is valid forH ∞ (R) atz=0, then the Myrberg phenomenon\(H^\infty (R) o \varphi = H^\infty (\tilde R)\) occurs. One might suspect that the converse also holds. In this paper, contrary to this intuition, we show that the converse of this previous result is not true. In addition, we generalize the previous result for more general sequences {c n}. By this generalization we can even partly simplify the previous proof.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. Forelli,A note on divisibility in H ∞ (X), Canad. J. Math.36 (1984), 458–469.

    MATH  MathSciNet  Google Scholar 

  2. M. Hayashi and M. Nakai,Point seperation by bounded analytic functions of a covering Riemann surface, Pacific J. Math.134 (1988), 261–273.

    MATH  MathSciNet  Google Scholar 

  3. M. Hayashi and M. Nakai,A uniqueness theorem and the Myrberg phenomenon, J. Analyse Math.76 (1998), 109–136.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Hayashi, M. Nakai and S. Segawa,Bounded analytic functions on two sheeted discs, Trans. Amer. Math. Soc.333 (1992), 799–819.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Hayashi, M. Nakai and S. Segawa,Two sheeted discs and bounded analytic functions, J. Analyse Math.61 (1993), 293–325.

    MATH  MathSciNet  Google Scholar 

  6. P. J. Myrberg,Über die Analytische Fortsetzung von beschränkten Funktionen, Ann. Acad. Sci. Fenn. Ser. A58 (1949), 1–7.

    Google Scholar 

  7. L. Zalcman,Bounded analytic functions on domains of infinite connectivity, Trans. Amer. Math. Soc.144 (1969), 241–269.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hokkaido University, 060-0810, Sapporo, Japan

    Mikihiro Hayashi & Yasuyuki Kobayashi

  2. Department of Mathematics, Nagoya Institute of Technology, Gokiso, Showa, 466-8555, Nagoya, Japan

    Mitsuru Nakai

Authors
  1. Mikihiro Hayashi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yasuyuki Kobayashi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mitsuru Nakai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikihiro Hayashi.

Additional information

To complete the present work the first and second (third, resp.) named authors were supported in part by Grant-in-Aid for Scientific Research, No. 10304010 (10640190, 11640187, resp.), Japanese Ministry of Education, Science and Culture.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hayashi, M., Kobayashi, Y. & Nakai, M. A uniqueness theorem and the myrberg phenomenon for a Zalcman domain. J. Anal. Math. 82, 267–283 (2000). https://doi.org/10.1007/BF02791230

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02791230

Keywords

Search

Navigation