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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Linear space properties of $ H^p$ spaces of Dirichlet series


Authors: Andriy Bondarenko, Ole Fredrik Brevig, Eero Saksman and Kristian Seip
Journal: Trans. Amer. Math. Soc. 372 (2019), 6677-6702
MSC (2010): Primary 30H10; Secondary 46E10, 30B50
DOI: https://doi.org/10.1090/tran/7898
Published electronically: July 30, 2019
MathSciNet review: 4024535
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Abstract: We study $ H^p$ spaces of Dirichlet series, called $ \mathcal {H}^p$, for the range $ 0<p< \infty $. We begin by showing that two natural ways to define $ \mathcal {H}^p$ coincide. We then proceed to study some linear space properties of $ \mathcal {H}^p$. More specifically, we study linear functionals generated by fractional primitives of the Riemann zeta function; our estimates rely on certain Hardy-Littlewood inequalities and display an interesting phenomenon, called contractive symmetry between $ \mathcal {H}^p$ and $ \mathcal {H}^{4/p}$, contrasting with the usual $ L^p$ duality. We next deduce general coefficient estimates, based on an interplay between the multiplicative structure of $ \mathcal {H}^p$ and certain new one variable bounds. Finally, we deduce general estimates for the norm of the partial sum operator $ \sum _{n=1}^\infty a_n n^{-s}\mapsto \sum _{n=1}^N a_n n^{-s}$ on $ {{\mathcal {H}}^p}$ with $ 0< p \le 1$, supplementing a classical result of Helson for the range $ 1<p<\infty $. The results for the coefficient estimates and for the partial sum operator exhibit the traditional schism between the ranges $ 1\le p \le \infty $ and $ 0<p<1$.


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Additional Information

Andriy Bondarenko
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
Email: andriybond@gmail.com

Ole Fredrik Brevig
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
Email: ole.brevig@ntnu.no

Eero Saksman
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway, and Department of Mathematics and Statistics, University of Helsinki, FI-00170 Helsinki, Finland
Email: eero.saksman@helsinki.fi

Kristian Seip
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
Email: kristian.seip@ntnu.no

DOI: https://doi.org/10.1090/tran/7898
Received by editor(s): January 19, 2018
Received by editor(s) in revised form: February 20, 2019
Published electronically: July 30, 2019
Additional Notes: The research of the first, second, and fourth authors was supported in part by Grant 227768 of the Research Council of Norway.
The third author’s research was supported in part by the Lars Onsager Professorship at NTNU, and in part by the Finnish Academy CoE “Analysis and Dynamics”.
Article copyright: © Copyright 2019 American Mathematical Society