Book contents
- Frontmatter
- Contents
- Introduction
- Part 1 Bohr’s Problem and Complex Analysis on Polydiscs
- 1 The Absolute Convergence Problem
- 2 Holomorphic Functions on Polydiscs
- 3 Bohr’s Vision
- 4 Solution to the Problem
- 5 The Fourier Analysis Point of View
- 6 Inequalities I
- 7 Probabilistic Tools I
- 8 Multidimensional Bohr Radii
- 9 Strips under the Microscope
- 10 Monomial Convergence of Holomorphic Functions
- 11 Hardy Spaces of Dirichlet Series
- 12 Bohr’s Problem in Hardy Spaces
- 13 Hardy Spaces and Holomorphy
- Part 2 Advanced Toolbox
- Part 3 Replacing Polydiscs by Other Balls
- Part 4 Vector-Valued Aspects
- References
- Symbol Index
- Subject Index
12 - Bohr’s Problem in Hardy Spaces
from Part 1 - Bohr’s Problem and Complex Analysis on Polydiscs
Published online by Cambridge University Press: 19 July 2019
- Frontmatter
- Contents
- Introduction
- Part 1 Bohr’s Problem and Complex Analysis on Polydiscs
- 1 The Absolute Convergence Problem
- 2 Holomorphic Functions on Polydiscs
- 3 Bohr’s Vision
- 4 Solution to the Problem
- 5 The Fourier Analysis Point of View
- 6 Inequalities I
- 7 Probabilistic Tools I
- 8 Multidimensional Bohr Radii
- 9 Strips under the Microscope
- 10 Monomial Convergence of Holomorphic Functions
- 11 Hardy Spaces of Dirichlet Series
- 12 Bohr’s Problem in Hardy Spaces
- 13 Hardy Spaces and Holomorphy
- Part 2 Advanced Toolbox
- Part 3 Replacing Polydiscs by Other Balls
- Part 4 Vector-Valued Aspects
- References
- Symbol Index
- Subject Index
Summary
Each Hardy space of Dirichlet series \mathcal{H}_p has an associated abscissa, and the analogue to Bohr’s problem arises in a natural way: to determine the maximal distance S_p between this abscissa and the abscissa of absolute convergence. If a Dirichlet series with coefficients (a_n) belongs to \mathcal{H}_p, then the series with coefficients (a_n/n^{ε}) belongs to \mathcal{H}_q for all q>p and ε >0. It is shown that S_p=1/2, and that, if we only consider m-homogeneous Dirichlet series, S_p^m=1/2. For every 1 ≤ p < ∞ the set of monomial convergence of the Hardy space H_p of functions on the infinite dimensional polytorus (hence also of the Hardy space H_2 on the infinite-dimensional polytorus) is l_2 ∩ Bc0. The space of all multipliers on the Hardy space of Dirichlet series \mathcal{H}_p coincides with \mathcal{H}_\infty.
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- Dirichlet Series and Holomorphic Functions in High Dimensions , pp. 289 - 314Publisher: Cambridge University PressPrint publication year: 2019