Advertisement

Interpolating Sequence for Multipliers of \(D_{\log }\) Space

  • Jizhen Zhou
  • Wei Chen
Original Paper
  • 7 Downloads

Abstract

In this paper, we investigate the Dirichlet type space \(D_{\log }\), which is closely associated with the analytic version of \(\mathcal Q_1(\partial {\mathbb {D}})\) space. We show that the space \(D_{\log }\) has the Pick Property. A characterization of interpolating sequence for multipliers of \(D_{\log }\) is given.

Keywords

Dirichlet type space Interpolating sequence Carleson measure 

Mathematics Subject Classification

Primary 30H25 Secondary 30D45 

Notes

Acknowledgements

The author thanks the referee for a careful reading of the paper and for providing a number of helpful suggestions and corrections. The paper is supported by NSF of Anhui Province, China (no. 1608085MA01)

References

  1. 1.
    Agler, J., MaCarthy, J.: Pick Interpolation and Hilbert Function Spaces. American Mathematical Society, Providence (2002)CrossRefGoogle Scholar
  2. 2.
    Arcozzi, N., Blasi, D., Pau, J.: Interpolating sequences on analytic Besov type spaces. Indiana Univ. Math. J. 58, 1281–1318 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aulaskari, R., Lappan, P.: Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal. In: Complex Analysis and Its Applications, Pitman Research Notes in Mathematics 305, Longman Scientific & Technical, Harlow, pp. 136–146 (1994)Google Scholar
  4. 4.
    Baernstein, A.: Analytic Functions of Bounded Mean Oscillation, Aspects of Contemporary Complex Analysis, pp. 3–36. Academic Press, Cambridge (1980)Google Scholar
  5. 5.
    Bao, G., Pau, J.: Boundary multipliers of a family of Mobius invariant function spaces. Ann. Acad. Sci. Fenn. Math. 41, 199–220 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bao, G., Lou, Z., Qian, R., Wulan, H.: On multipliers of Dirichlet type spaces. Complex Anal. Oper. Theory 9, 1701–1732 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Böe, B.: Interpolating sequences for Besov spaces. J. Funct. Anal. 192, 319–341 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Böe, B.: An interpolation theorem for Hilbert spaces with Nevanlinna–Pick kernel. Proc. Am. Math. Soc. 133, 2077–2081 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Böe, B., Nicolau, A.: Interpolation by functions in the Bloch space. J. Anal. Math. 94, 171–194 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Carleson, L.: An interpolations problem for bounded analytic functions. Am. J. Math. 80, 921–930 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Essen, M., Wulan, H., Xiao, J.: Several function-theoretic characterizations of Möbious invariant \({\cal{Q}}_K\) spaces. J. Funct. Anal. 230, 78–115 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Essén, M., Xiao, J.: Some results on \({\cal{Q}}_p\) spaces, \(0<p<1\). J. Reine Angew. Math. 485, 173–195 (1997)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Garnett, J.: Bounded Analytic Functions. Academic Press, New York (1981)zbMATHGoogle Scholar
  14. 14.
    Hardy, G.: Divergent Series. Claredon Press, New York (1949)zbMATHGoogle Scholar
  15. 15.
    Marshall, D., Sunderg, C.: Interpolation sequences for the multipliers of the Dirichler space. Manuscript (1994). https://www.math.washington.edu/~marshall/preprints/interp. Accessed 13 July 2018
  16. 16.
    Maz’ya, D., Shaposhnikova, T.: Theory of multipliers in spaces of differentiable functions. Russ. Math. Surv. 38, 23–95 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Schuster, A.: Interpolation by Bloch functions. Ill. J. Math. 43, 677–691 (1999)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Seip, K.: Interpolation and Sampling in Spaces of Analytic Functions, University Lecture Series, 33. American Mathematical Society, Providence (2004)Google Scholar
  19. 19.
    Wulan, H., Zhou, J.: Decomposition theorem and discreteness form for \({\cal{Q}}\_{K}\) spaces. Forum Math. 26(214), 467–495 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Wulan, H., Zhu, K.: Möbius Invariant \({\cal{Q}}\_K\) Spaces. Springer International Publishing, New York (2017)CrossRefzbMATHGoogle Scholar
  21. 21.
    Xiao, J.: Some essential properties of \({\cal{Q}}\_p(\partial \Delta )\)-spaces. J. Fourier Anal. Appl. 6, 311–323 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Xiao, J.: Holomorphic \({\cal{Q}}\) Classes. Springer, Berlin (2001). (LNM 1767) CrossRefGoogle Scholar
  23. 23.
    Xiao, J.: Geometric \({\cal{Q}}\_p\) Functions. Birkhäuser Verlag, Basel (2006)Google Scholar
  24. 24.
    Zhou, J., Bao, G.: Analytic version of \({\cal{Q}}_1(\partial {\mathbb{D}})\) space. J. Math. Anal. Appl. 422, 1091–1102 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zhou, J., Wu, Y.: Decomposition theorems and conjugate pair in \(D\_K\) spaces. Acta Math. Sin. (Chin. Ser.) 30, 1513–1525 (2014)CrossRefzbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.School of Mathematics and Big DataAnhui University of Science and TechnologyHuainanChina

Personalised recommendations