Linear space properties of $H^p$ spaces of Dirichlet series
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- by Andriy Bondarenko, Ole Fredrik Brevig, Eero Saksman and Kristian Seip PDF
- Trans. Amer. Math. Soc. 372 (2019), 6677-6702 Request permission
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 HardyâLittlewood 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 $0<p<1$.References
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Additional Information
- Andriy Bondarenko
- Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
- MR Author ID: 763910
- Email: andriybond@gmail.com
- Ole Fredrik Brevig
- Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
- MR Author ID: 1069722
- 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
- MR Author ID: 315983
- Email: eero.saksman@helsinki.fi
- Kristian Seip
- Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
- MR Author ID: 158300
- Email: kristian.seip@ntnu.no
- 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 authorâs research was supported in part by the Lars Onsager Professorship at NTNU, and in part by the Finnish Academy CoE âAnalysis and Dynamicsâ. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 6677-6702
- MSC (2010): Primary 30H10; Secondary 46E10, 30B50
- DOI: https://doi.org/10.1090/tran/7898
- MathSciNet review: 4024535