Monatshefte für Mathematik

, Volume 136, Issue 3, pp 203–236

Hardy Spaces of Dirichlet Series and Their Composition Operators

  • Frédéric Bayart

DOI: 10.1007/s00605-002-0470-7

Cite this article as:
Bayart, F. Mh Math (2002) 136: 203. doi:10.1007/s00605-002-0470-7


 In [9], Hedenmalm, Lindqvist and Seip introduce the Hilbert space of Dirichlet series with square summable coefficients \(\), and begin its study, with modern functional and harmonic analysis tools. The space \(\) is an analogue for Dirichlet series of the space \(\) for Fourier series. We continue their study by introducing \(\), an analogue to the spaces \(\). Thanks to Bohr’s vision of Dirichlet series, we identify \(\) with the Hardy space of the infinite polydisk \(\). Next, we study a variant of the Poisson semigroup for Dirichlet series. We give a result similar to the one of Weissler ([25]) about the hypercontractivity of this semigroup on the spaces \(\). Finally, following [8], we determine the composition operators on \(\), and we compare some properties of such an operator and of its symbol.

2000 Mathematics Subject Classification: 42B30 47B33 32A55 
Key words: Dirichlet series hypercontractivity composition operators Bohr Hardy spaces infinite polydisc 

Copyright information

© Springer-Verlag Wien 2002

Authors and Affiliations

  • Frédéric Bayart
    • 1
  1. 1.Université des Sciences et Technologies de Lille, Villeneuve d’Ascq, FranceFR

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