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Optimal comparison of the p-norms of Dirichlet polynomials

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Abstract

Let 1 ≤ p < q < ∞. We show that

$$\sup \frac{{{{\left\| D \right\|}_{{H_q}}}}}{{{{\left\| D \right\|}_{{H_q}}}}} = \exp \left( {\frac{{\log x}}{{\log \log x}}\left( {\log \sqrt {\frac{q}{p}} + O\left( {\frac{{\log \log \log x}}{{\log \log x}}} \right)} \right)} \right),$$

where the supremum is taken over all non-zero Dirichlet polynomials of the form D(s) = Σ n≤x a n n −s and

$${\left\| D \right\|_{{H_q}}} = \mathop {\lim }\limits_{T \to \infty } {\left( {\frac{1}{{2T}}\int_{ - T}^T | \sum\limits_{n \leqslant x} {{a_n}{n^{ - it}}\left| {^pdt} \right.} } \right)^{1/p}}.$$

An application is given to the study of multipliers between Hardy spaces H p of Dirichlet series.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. F. Bayart, H. Queffélec and K. Seip, Approximation numbers of composition operators on Hp spaces of Dirichlet series, Ann. Inst. Fourier (Grenoble) 66 (2016), 551–588.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. S. Besicovitch, On Generalized Almost Periodic Functions, Proc. London Math. Soc. S2-25 (1926), 495–512.

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Bohr, Über die bedeutung der potenzreihen unendlich vieler variabeln in der theorie der dirichletschen reihen a n/n s, Nachr. Ges. Wiss. Gött. Math. Phys. Kl. 4 (1913), 441–488.

    MATH  Google Scholar 

  5. O. F. Brevig, On the Sidon constant for Dirichlet polynomials, Bull. Sci. Math. 138 (2014), 656–664.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Carando, A. Defant and P. Sevilla-Peris, Some polynomial versions of cotype and applications, J. Funct. Anal. 270 (2016), 68–87.

    Article  MathSciNet  MATH  Google Scholar 

  7. B. J. Cole and T. W. Gamelin, Representing measures and Hardy spaces for the infinite polydisk algebra, Proc. London Math. Soc. (3) 53 (1986), 112–142.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. de la Bretèche, Sur l’ordre de grandeur des polynomes de Dirichlet, Acta Arith. 134 (2008), 141–148.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Defant, L. Frerick, J. Ortega-Cerdà, M. Ounäies and K. Seip, The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. (2) 174 (2011), 485–497.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Defant and M. Mastylo, Lp-norms and Mahler’s measure of polynomials on the ndimensional torus, Constr. Approx. 44 (2016), 87–101.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Dursat, Autour de la fonction qui compte le nombre de nombres premiers, Thèse, Université de Limoges, 1998, available online: http://www.unilim.fr/laco/theses/1998/T1998_01.pdf.

    Google Scholar 

  12. H. Hedenmalm, P. Lindqvist and K. Seip, A Hilbert space of Dirichlet series and systems of dilated functions in L2(0, 1), Duke Math. J. 86 (1997), 1–37.

    Article  MathSciNet  MATH  Google Scholar 

  13. H. Helson, Hankel forms and sums of random variables, Studia Math. 176 (2006), 85–92.

    Article  MathSciNet  MATH  Google Scholar 

  14. H. König and S. Kwapien, Best Khintchine type inequalities for sums of independent, rotationally invariant random vectors, Positivity 5 (2001), 115–152.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. V. Konyagin and H. Queffélec, The translation 1/2 in the theory of Dirichlet series, Real Anal. Exchange 27 (2001/02), 155–175.

    MathSciNet  MATH  Google Scholar 

  16. H. Queffélec and M. Queffélec, Diophantine approximation and Dirichlet series, Harish-Chandra Research Institute Lecture Notes, Vol. 2, Hindustan Book Agency, New Delhi, 2013.

    Google Scholar 

  17. E. Saksman and K. Seip, Some open questions in analysis for Dirichlet series, in Recent progress on operator theory and approximation in spaces of analytic functions, Contemp. Math., Vol. 679, Amer. Math. Soc., Providence, RI, 2016, pp. 179–191.

    Chapter  Google Scholar 

  18. G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics, Vol. 46, Cambridge University Press, Cambridge, 1995.

    Google Scholar 

  19. F. B. Weissler, Logarithmic Sobolev inequalities and hypercontractive estimates on the circle, J. Funct. Anal. 37 (1980), 218–234.

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Wigert, Sur l’ordre de grandeur du nombre des diviseurs d’un entier, Arkiv. für Math. 3 (1907), 1–9.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für mathematik, Universität Oldenburg, D-26111, Oldenburg, Germany

    A. Defant

  2. Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100, Murcia, Spain

    A. Pérez

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  1. A. Defant
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  2. A. Pérez
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Correspondence to A. Pérez.

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Defant, A., Pérez, A. Optimal comparison of the p-norms of Dirichlet polynomials. Isr. J. Math. 221, 837–852 (2017). https://doi.org/10.1007/s11856-017-1576-x

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  • DOI: https://doi.org/10.1007/s11856-017-1576-x

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