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  • Print publication year: 2019
  • Online publication date: July 2019

12 - Bohr’s Problem in Hardy Spaces

from Part 1 - Bohr’s Problem and Complex Analysis on Polydiscs

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.