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Convergence and almost sure properties in Hardy spaces of Dirichlet series

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Abstract

Given a frequency \(\lambda \), we study general Dirichlet series \(\sum a_n e^{-\lambda _n s}\). First, we give a new condition on \(\lambda \) which ensures that a somewhere convergent Dirichlet series defining a bounded holomorphic function in the right half-plane converges uniformly in this half-plane, improving classical results of Bohr and Landau. Then, following recent works of Defant and Schoolmann, we investigate Hardy spaces of these Dirichlet series. We get general results on almost sure convergence which have an harmonic analysis flavour. Nevertheless, we also exhibit examples showing that it seems hard to get general results on these spaces as spaces of holomorphic functions.

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References

  1. Bayart, F.: Hardy spaces of Dirichlet series and their composition operators. Monatsh. Math. 136, 203–236 (2002)

    Article  MathSciNet  Google Scholar 

  2. Bayart, F., Heurteaux, Y.: Multifractal analysis of the divergence of Fourier series, the extreme cases. J. Anal. Math. 124, 387–408 (2014)

    Article  MathSciNet  Google Scholar 

  3. Bayart, F., Mastylo, M.: Interpolation of Hardy spaces of Dirichlet series]. J. Funct. Anal. 277(3), 786–805 (2019)

    Article  MathSciNet  Google Scholar 

  4. Bayart, F., Queffélec, H., Seip, K.: Approximation numbers of composition operators on \(H^p\) spaces of Dirichlet series. Annales de l’Institut Fourier 66, 551–588 (2016)

    Article  MathSciNet  Google Scholar 

  5. Bohr, H.: Über die gleichmäßige Konvergenz Dirichletscher Reihen. J. Reine Angew. Math. 143, 203–211 (1913)

    Article  MathSciNet  Google Scholar 

  6. Bondarenko, A., Brevig, O., Saksman, E., Seip, K.: Linear properties of \(H^p\) spaces of Dirichlet series. Trans. Amer. Math. Soc. 372, 6677–6702 (2019)

    Article  MathSciNet  Google Scholar 

  7. Choi, Y.S., Kim, U.Y., Maestre, M.: Banach spaces of general Dirichlet series. J. Math. Anal. Appl. 465, 839–856 (2018)

    Article  MathSciNet  Google Scholar 

  8. Defant, A., Pérez, A.: Optimal comparison of the \(p\)-norms of Dirichlet polynomials. Israel J. Math. 221, 837–852 (2017)

    Article  MathSciNet  Google Scholar 

  9. Defant, A., Schoolmann, I.: Hardy spaces of general Dirichlet series - a survey. Function Spaces XII, vol. 119, Banach Center Publications, pp. 123–149 (2019)

  10. Defant, A., Schoolmann, I.: \(\cal{H}_p\)-theory of general Dirichlet series. J. Fourier Anal. Appl. 25, 3220–3258 (2019)

    Article  MathSciNet  Google Scholar 

  11. Defant, A., Schoolmann, I.: Riesz means in Hardy spaces on Dirichlet groups. Math. Ann. 378, 57–96 (2020). arXiv:1908.06458

  12. Defant, A., Schoolmann, I.: Variants of a theorem of Helson for general Dirichlet series. J. Funct. Anal. 279, 108569 (2020). arXiv:1909.11990

    Article  MathSciNet  Google Scholar 

  13. Grafakos, L.: Modern Fourier analysis, Graduate texts in mathematics, vol. 250. Springer, New York (2014)

    Google Scholar 

  14. Hardy, G.H., Riesz, M.: The general theory of Dirichlet series, vol. 18. Cambridge University Press, Cambridge (1915)

    MATH  Google Scholar 

  15. Hedenmalm, H., Lindqvist, P., Seip, K.: A Hilbert space of Dirichlet series and a system of dilated functions in \(L^2([0,1])\). Duke Math. J. 86, 1–36 (1997)

    Article  MathSciNet  Google Scholar 

  16. Helson, H.: Conjugate series and a theorem of Paley. Pacific J. Math. 8, 437–446 (1958)

    Article  MathSciNet  Google Scholar 

  17. Helson, H.: Compact groups and Dirichlet series. Ark. Mat. 8, 139–143 (1969)

    Article  MathSciNet  Google Scholar 

  18. Helson, H.: Dirichlet series. Regent Press, California (2005)

    MATH  Google Scholar 

  19. Landau, E.: Über die gleichmäßige Konvergenz Dirichletscher Reihen. Math. Zeit. 11, 317–318 (1921)

    Article  MathSciNet  Google Scholar 

  20. Queffélec, H.: Espaces de séries de Dirichlet et leurs opérateurs de composition. Annales Math. Blaise Pascal 22, 267–344 (2015)

    Article  MathSciNet  Google Scholar 

  21. Schoolmann, I.: On Bohr’s theorem for general Dirichlet series, Math. Nach. 293, 1591–1612 (2020). arXiv:1812.04925

  22. Zygmund, A.: Trigonometric series, 2nd, vol. I and II combined, Cambridge University Press, London and New York, (1969). Two volumes bound as one, reprinted with corrections and some additions. 2nd edition volume I and II bound together. MR:0236587

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  1. Laboratoire de Mathématiques Blaise Pascal UMR 6620 CNRS, Université Clermont Auvergne, Campus universitaire des Cézeaux, 3 place Vasarely, 63178, Aubière Cedex, France

    Frédéric Bayart

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  1. Frédéric Bayart
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Correspondence to Frédéric Bayart.

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Communicated by Loukas Grafakos.

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The author was partially supported by the grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front).

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Bayart, F. Convergence and almost sure properties in Hardy spaces of Dirichlet series. Math. Ann. 382, 1485–1515 (2022). https://doi.org/10.1007/s00208-021-02239-x

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  • DOI: https://doi.org/10.1007/s00208-021-02239-x

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