Journal d'Analyse Mathématique

, Volume 103, Issue 1, pp 197–219

Interpolation and harmonic majorants in big Hardy-Orlicz spaces

Article

Abstract

Free interpolation in Hardy spaces is characterized by the well-known Carleson condition. The result extends to Hardy-Orlicz spaces contained in the scale of classical Hardy spaces Hp, p > 0. For the Smirnov and the Nevanlinna classes, interpolating sequences have been characterized in a recent paper in terms of the existence of harmonic majorants (quasi-bounded in the case of the Smirnov class). Since the Smirnov class can be regarded as the union over all Hardy-Orlicz spaces associated with a so-called strongly convex function, it is natural to ask how the condition changes from the Carleson condition in classical Hardy spaces to harmonic majorants in the Smirnov class. The aim of this paper is to narrow down this gap from the Smirnov class to “big” Hardy-Orlicz spaces. More precisely, we characterize interpolating sequences for a class of Hardy-Orlicz spaces that carry an algebraic structure and are strictly bigger than ⋃p>0Hp. It turns out that the interpolating sequences are again characterized by the existence of quasi-bounded majorants, but now the functions defining these quasi-bounded majorants have to be in suitable Orlicz spaces. The existence of harmonic majorants defined by functions in such Orlicz spaces is also discussed in the general situation. We finish the paper with a class of examples of separated Blaschke sequences which are interpolating for certain Hardy-Orlicz spaces without being interpolating for slightly smaller ones.

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References

  1. [Ca58]
    L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921–930.MATHCrossRefMathSciNetGoogle Scholar
  2. [Gar77]
    J. B. Garnett, Two remarks on interpolation by bounded analytic functions, in Banach Spaces of Analytic Functions, Springer Lecture Notes in Math. 604, 1977, pp. 32–40.Google Scholar
  3. [Gar81]
    J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.MATHGoogle Scholar
  4. [Har99]
    A. Hartmann, Free interpolation in Hardy-Orlicz spaces, Studia Math. 135 (1999), 179–190.MATHMathSciNetGoogle Scholar
  5. [HaMa01]
    A. Hartmann and X. Massaneda, Interpolating sequences for holomorphic functions of restricted growth, Illinois J. Math. 46 (2002), 929–945.MATHMathSciNetGoogle Scholar
  6. [HMNT04]
    A. Hartmann, X. Massaneda, P.J. Thomas and A. Nicolau, Interpolation in the Nevanlinna and Smirnov classes and harmonic majorants, J. Funct. Anal. 217 (2004), 1–37.MATHCrossRefMathSciNetGoogle Scholar
  7. [He69]
    M. Heins, Hardy Classes on Riemann Surfaces, Springer Lectures Notes in Mathematics 98, 1969.Google Scholar
  8. [Ka63]
    V. Kabaila, Interpolation sequences for the H p classes in the case p < 1, Litovsk. Mat. Sb. 3 (1963), 141–147.MATHMathSciNetGoogle Scholar
  9. [KrRu61]
    M. A. Krasnosel’skii and Ya. Rutickii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961.Google Scholar
  10. [Leś73]
    R. Leśniewicz, On linear functionals in Hardy-Orlicz spaces, I, Studia Math. 46 (1973), 53–77.MATHGoogle Scholar
  11. [Na56]
    A. G. Naftalevič, On interpolation by functions of bounded characteristic (Russian), Vilniaus Valst. Univ. Mokslu Darbai. Mat. Fiz. Chem. Mokslu Ser. 5 (1956), 5–27.MathSciNetGoogle Scholar
  12. [NPT]
    A. Nicolau, J. Pau and P.J. Thomas, Smallness sets for bounded holomorphic functions, J. Anal. Math. 82 (2000), 119–148.MATHMathSciNetCrossRefGoogle Scholar
  13. [Nik86]
    N. K. Nikol’skii, Treatise on the Shift Operator, Springer-Verlag, Berlin, 1986.MATHGoogle Scholar
  14. [Nik02]
    N.K. Nikol’skii, Operators, Functions, and Systems: An Easy Reading. Vol. 1, Hardy, Hankel, and Toeplitz; Vol.2, Model Operators and Systems, American Mathematical Society, Providence, RI, 2002.Google Scholar
  15. [Pe67]
    A.L. Peressini, Ordered Topological Vector Spaces, Harper & Row, New York-London, 1967.MATHGoogle Scholar
  16. [RosRov85]
    M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, The Clarendon Press, Oxford University Press, New York, 1985.MATHGoogle Scholar
  17. [ShHSh]
    H.S. Shapiro and A.L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513–532.MATHCrossRefMathSciNetGoogle Scholar
  18. [Ya74]
    N. Yanagihara, Interpolation theorems for the class N +, Illinois J. Math. 18 (1974), 427–435.MATHMathSciNetGoogle Scholar

Copyright information

© The Hebrew University of Jerusalem 2007

Authors and Affiliations

  1. 1.Laboratoire Bordelais d’Analyse et GéométrieUniversité Bordeaux ITalenceFrance

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