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Interpolation and harmonic majorants in big Hardy-Orlicz spaces

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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 H p, 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>0 H p. 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. L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921–930.

    Article  MATH  MathSciNet  Google Scholar 

  2. 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.

  3. J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.

    MATH  Google Scholar 

  4. A. Hartmann, Free interpolation in Hardy-Orlicz spaces, Studia Math. 135 (1999), 179–190.

    MATH  MathSciNet  Google Scholar 

  5. A. Hartmann and X. Massaneda, Interpolating sequences for holomorphic functions of restricted growth, Illinois J. Math. 46 (2002), 929–945.

    MATH  MathSciNet  Google Scholar 

  6. 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.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Heins, Hardy Classes on Riemann Surfaces, Springer Lectures Notes in Mathematics 98, 1969.

  8. V. Kabaila, Interpolation sequences for the H p classes in the case p < 1, Litovsk. Mat. Sb. 3 (1963), 141–147.

    MATH  MathSciNet  Google Scholar 

  9. M. A. Krasnosel’skii and Ya. Rutickii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961.

    Google Scholar 

  10. R. Leśniewicz, On linear functionals in Hardy-Orlicz spaces, I, Studia Math. 46 (1973), 53–77.

    MATH  Google Scholar 

  11. 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.

    MathSciNet  Google Scholar 

  12. A. Nicolau, J. Pau and P.J. Thomas, Smallness sets for bounded holomorphic functions, J. Anal. Math. 82 (2000), 119–148.

    Article  MATH  MathSciNet  Google Scholar 

  13. N. K. Nikol’skii, Treatise on the Shift Operator, Springer-Verlag, Berlin, 1986.

    MATH  Google Scholar 

  14. 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. A.L. Peressini, Ordered Topological Vector Spaces, Harper & Row, New York-London, 1967.

    MATH  Google Scholar 

  16. M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, The Clarendon Press, Oxford University Press, New York, 1985.

    MATH  Google Scholar 

  17. H.S. Shapiro and A.L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513–532.

    Article  MATH  MathSciNet  Google Scholar 

  18. N. Yanagihara, Interpolation theorems for the class N +, Illinois J. Math. 18 (1974), 427–435.

    MATH  MathSciNet  Google Scholar 

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  1. Laboratoire Bordelais d’Analyse et Géométrie, Université Bordeaux I, 351 cours de la Libération, 33405, Talence, France

    Andreas Hartmann

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  1. Andreas Hartmann
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Correspondence to Andreas Hartmann.

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Hartmann, A. Interpolation and harmonic majorants in big Hardy-Orlicz spaces. J Anal Math 103, 197–219 (2007). https://doi.org/10.1007/s11854-008-0006-8

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  • DOI: https://doi.org/10.1007/s11854-008-0006-8

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