Blaschke Products and Zero Sets in Weighted Dirichlet Spaces

Abstract

In this paper, we deal with superharmonically weighted Dirichlet spaces \(\mathcal {D}_{\omega }\). First, we prove that the classical Dirichlet space is the largest, among all these spaces, which contains no infinite Blaschke product. Next, we give new sufficient conditions on a Blaschke sequence to be a zero set for \(\mathcal {D}_{\omega }\). Our conditions improve Shapiro-Shields condition for \(\mathcal {D}_{\alpha }\), when α ∈ (0,1).

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Agler, J., McCarthy, J.E.: Pick interpolation and Hilbert function spaces, vol. 44 of graduate studies in mathematics. American Mathematical Society, Providence (2002)

    Google Scholar 

  2. 2.

    Aleman, A.: Hilbert spaces of analytic functions between the Hardy and the Dirichlet space. Proc. Amer. Math. Soc. 115(1), 97–104 (1992)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Aleman, A.: The multiplication operator on Hilbert spaces of analytic functions. Habilitationsschrift, Fern Universitat (1993)

    Google Scholar 

  4. 4.

    Aleman, A., Vukotić, D.: On Blaschke products with derivatives in Bergman spaces with normal weights. J. Math. Anal. Appl. 361(2), 492–505 (2010)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Armitage, D.H., Gardiner, S.J.: Classical potential theory. Springer Monographs in Mathematics. Springer-Verlag, London (2001)

    Book  Google Scholar 

  6. 6.

    Bao, G., Göğüş, N., Pouliasis, S.: On Dirichlet spaces with a class of superharmonic weights. Canad. J. Math. 70(4), 721–741 (2018)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Beurling, A., Deny, J.: Espaces de Dirichlet. I. Le cas élémentaire. Acta Math. 99, 203–224 (1958)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bourhim, A., El-Fallah, O., Kellay, K.: Boundary behaviour of functions of Nevanlinna class. Indiana Univ. Math J. 53(2), 347–395 (2004)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Carleson, L.: On a class of meromorphic functions and its associated exceptional sets. Uppsala (1950)

  10. 10.

    Carleson, L.: On the zeros of functions with bounded Dirichlet integrals. Math. Z. 56, 289–295 (1952)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Carleson, L.: Sets of uniqueness for functions regular in the unit circle. Acta Math. 87, 325–345 (1952)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Duren, P.L.: Theory of Hp spaces. Pure and applied mathematics, vol. 38. Academic Press, New York-London (1970)

    Google Scholar 

  13. 13.

    Dyakonov, K.M.: Factorization of smooth analytic functions, and Hilbert-Schmidt operators. Algebra i Analiz 8(4), 1–42 (1996)

    MathSciNet  Google Scholar 

  14. 14.

    El-Fallah, O., Elmadani, Y., Kellay, K.: Cyclicity and invariant subspaces in Dirichlet spaces. J. Funct. Anal. 270(9), 3262–3279 (2016)

    MathSciNet  Article  Google Scholar 

  15. 15.

    El-Fallah, O., Elmadani, Y., Kellay, K.: Kernel and capacity estimates in Dirichlet spaces. J. Funct. Anal. 276(3), 867–895 (2019)

    MathSciNet  Article  Google Scholar 

  16. 16.

    El-Fallah, O., Kellay, K., Klaja, H., Mashreghi, J., Ransford, T.: Dirichlet spaces with superharmonic weights and de Branges–Rovnyak spaces. Complex Anal. Oper. Theory 10(1), 97–107 (2016)

    MathSciNet  Article  Google Scholar 

  17. 17.

    El-Fallah, O., Kellay, K., Mashreghi, J., Ransford, T.: A primer on the Dirichlet space, vol. 203 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2014)

    MATH  Google Scholar 

  18. 18.

    El-Fallah, O., Kellay, K., Ransford, T.: Cyclicity in the Dirichlet space. Ark Mat. 44(1), 61–86 (2006)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes, extended ed., vol. 19 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin (2011)

    Google Scholar 

  20. 20.

    Garnett, J.B.: Bounded analytic functions, first ed., vol. 236 of Graduate Texts in Mathematics. Springer, New York (2007)

    Google Scholar 

  21. 21.

    Guillot, D.: Blaschke condition and zero sets in weighted Dirichlet spaces. Ark. Mat. 50(2), 269–278 (2012)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Kellay, K., Mashreghi, J.: On zero sets in the Dirichlet space. J. Geom. Anal. 22(4), 1055–1070 (2012)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Marshall, D.E., Sundberg, C.: Interpolating sequences for the multipliers of the Dirichlet space (1994)

  24. 24.

    Nagel, A., Rudin, W., Shapiro, J.H.: Tangential boundary behavior of functions in Dirichlet-type spaces. Ann. of Math. (2) 116(2), 331–360 (1982)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Pau, J., Peláez, J.A.: On the zeros of functions in Dirichlet-type spaces. Trans. Amer. Math. Soc. 363(4), 1981–2002 (2011)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Pérez-González, F., Rättyä, J., Reijonen, A.: Derivatives of inner functions in Bergman spaces induced by doubling weights. Ann. Acad. Sci. Fenn Math. 42(2), 735–753 (2017)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Ransford, T.: Potential theory in the complex plane, vol. 28 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  28. 28.

    Richter, S., Sundberg, C.: A formula for the local Dirichlet integral. Michigan Math. J. 38(3), 355–379 (1991)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Richter, S., Yilmaz, F.: Regularity for generators of invariant subspaces of the Dirichlet shift. J. Funct Anal. 277(7), 2117–2132 (2018)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Seip, K.: Interpolation and sampling in spaces of analytic functions, vol. 33 of University Lecture Series. American Mathematical Society, Providence (2004)

    Google Scholar 

  31. 31.

    Shapiro, H.S., Shields, A.L.: On the zeros of functions with finite Dirichlet integral and some related function spaces. Math. Z. 80, 217–229 (1962)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Shimorin, S.: Complete Nevanlinna-Pick property of Dirichlet-type spaces. J. Funct. Anal. 191(2), 276–296 (2002)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Taylor, B.A., Williams, D.L.: Ideals in rings of analytic functions with smooth boundary values. Canad. J Math. 22, 1266–1283 (1970)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Verbitskiı̆, I.E.: Inner functions, Besov spaces and multipliers. Dokl. Akad. Nauk. SSSR 276(1), 11–14 (1984)

    MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the referee for his valuable remarks and suggestions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to O. El-Fallah.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Research partially supported by “Hassan II Academy of Science and Technology”.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Idrissi, H.BE., El-Fallah, O. Blaschke Products and Zero Sets in Weighted Dirichlet Spaces. Potential Anal 53, 1299–1316 (2020). https://doi.org/10.1007/s11118-019-09807-6

Download citation

Keywords

  • Blaschke product
  • Dirichlet space
  • Capacity

Mathematics Subject Classification (2010)

  • 31C25
  • 30J10
  • 31C15