Hardy spaces of general Dirichlet series — a survey

Abstract

The main purpose of this article is to survey on some key elements of a recent H p -theory of general Dirichlet series a n e −λns , which was mainly inspired by the work of Bayart and Helson on ordinary Dirichlet series a n n −s . In view of an ingenious identification of Bohr, the H p -theory of ordinary Dirichlet series can be seen as a sub-theory of Fourier analysis on the infinite dimensional torus T ∞ . Extending these ideas, the H p -theory of λ-Dirichlet series is build as a sub-theory of Fourier analysis on what we call λ-Dirichlet groups. A number of problems is added.

Fréchet spaces of general Dirichlet series

“…One of our main tools is the representation of our pre-Fréchet spaces as countable projective limits of their natural 'Banach space precursors' (D ∞ (λ ) and H p (λ )). In this sense our article continues a series of recent articles on general Dirichlet series (see [8,11,14,26,25]), which combine classical results from the deep analysis presented by Hardy and Riesz in [17], with various topics from modern analysis (as complex analysis, functional analysis in Banach and Fréchet spaces, Fourier analysis on R, or har-monic analysis on compact abelian groups).…”

“…But in general (D ∞ (λ ), r ∞ ) is no Banach space, or equivalently it does not form a closed subspace of [26,Theorem 5.2]). The normed space D ∞ (λ ), and in particular the question when it is complete, was extensively studied in [8,11,14,26,25].…”

Hausdorff Operators on Compact Abelian Groups

“…To formulate and prove similar results on general Dirichlet series we need some notation, definitions, and results from [6], [29], and [7].…”

“…This observation and especially the seminal result of Hedenmalm, Lindqvist, and Seip [12] give us an opportunity for an application of our results to ordinary Dirichlet series. In the case of general Dirichlet series we use similar results obtained by Defant and Schoolman in [6], [29], and [7]. Based on this results, classes of bounded Hausdorff operators that act in some classical spaces of Dirichlet series (ordinary and general) are introduced.…”

Vector-valued general Dirichlet series

“…The spaces H p (λ, X). From [28] (see also [31]) we recall the definition of and some basic facts about Dirichlet groups and we refer to [57] for background on Fourier analysis on groups. Let G be a compact abelian group and β : (R, +) → G a homomorphism of groups.…”