Hardy spaces of general Dirichlet series - a survey

TL;DR

This survey reviews recent developments in the $_p$-theory of general Dirichlet series, extending classical Fourier analysis techniques to broader settings involving $$-Dirichlet groups.

Contribution

It synthesizes recent advances in $_p$-theory for Dirichlet series and introduces the framework of $$-Dirichlet groups for analysis.

Findings

Extension of $_p$-theory to general Dirichlet series

Identification of $_p$-theory as a sub-theory of Fourier analysis on $$-Dirichlet groups

Discussion of open problems in the field

Abstract

The main purpose of this article is to survey on some key elements of a recent $\mathcal{H}_p$-theory of general Dirichlet series $\sum a_n e^{-\lambda_{n}s}$, which was mainly inspired by the work of Bayart and Helson on ordinary Dirichlet series $\sum a_n n^{-s}$. In view of an ingenious identification of Bohr, the $\mathcal{H}_p$-theory of ordinary Dirichlet series can be seen as a sub-theory of Fourier analysis on the infinite dimensional torus $\mathbb{T}^\infty$. Extending these ideas, the $\mathcal{H}_p$-theory of $\lambda$-Dirichlet series is build as a sub-theory of Fourier analysis on what we call $\lambda$-Dirichlet groups. A number of problems is added.

Equations

$${\sigma }_c(D)\le {\sigma }_b(D)\le {\sigma }_u(D)\le {\sigma }_a(D).$$

$${\sigma }_b(D)={\sigma }_u(D),$$

$$S=\sup \{{\sigma }_a(D)-{\sigma }_u(D):D=\sum a_n{n}^{-s}\},$$

$$S=\frac{1}{2}$$

$$\hat{f}(\alpha )={\int }_{{\mathbb{T}}^{\infty }}f(w)w^{-\alpha }dw=0,$$

$$H_{\infty }({\mathbb{T}}^{\infty })={\mathcal{D}}_{\infty },$$

$$\hat{f}(\alpha )=a_n\text{ whenever }n={\mathfrak{p}}^{\alpha }:=p_1^{{\alpha }_1}\dots p_N^{{\alpha }_N},$$

$$\beta :\mathbb{R}\to {\mathbb{T}}^{\infty },t\mapsto (p_k^{-it}{)}_{k=1}^{\infty }$$

$${\mathbb{T}}^{\infty }$$

$$H_{\infty }({\mathbb{T}}^{\infty })=H_{\infty }(B_{c_0}),\text{leavevmode}f\mapsto g$$

$$\hat{f}(\alpha )=\frac{{\partial }^{\alpha }g(0)}{\alpha !}$$

$${\mathcal{H}}_p=H_p({\mathbb{T}}^{\infty }),$$

$${\mathcal{D}}_{\infty }={\mathcal{H}}_{\infty }=H_{\infty }({\mathbb{T}}^{\infty }).$$

$$H_p({\ell }_2\cap B_{c_0})$$

$$\Vert g{\Vert }_{H_p({\ell }_2\cap B_{c_0})}=\underset{n\in \mathbb{N}}{\sup }\underset{0<r<1}{\sup }{\left({\int }_{{\mathbb{T}}^n}|g(r{w}_1,\dots ,r{w}_n,0,0,\dots ){|}^{p}d(w_1,\dots ,w_n)\right)}^{\frac{1}{p}}<\infty .$$

$$H_p({\ell }_2\cap B_{c_0})=H_p({\mathbb{T}}^{\infty }).$$

$$\sum a_n{e}^{-{\lambda }_{n}s}$$

$${\sigma }_c(D)=\inf \left\{\sigma \in \mathbb{R}\mid D\text{ converges on }[Re>\sigma ]\right\},$$

$${\sigma }_a(D)=\inf \left\{\sigma \in \mathbb{R}\mid D\text{ converges absolutely on }[Re>\sigma ]\right\},$$

$${\sigma }_u(D)=\inf \left\{\sigma \in \mathbb{R}\mid D\text{ converges uniformly on }[Re>\sigma ]\right\},$$

$$\sigma_{c}(D)\leq\limsup_{N}\frac{\log\Big{(}\big{|}\sum_{n=1}^{N}a_{n}\big{|}\Big{)}}{\lambda_{N}}\leavevmode\nobreak\ \text{ and }\leavevmode\nobreak\ \sigma_{a}(D)\leq\limsup_{N}\frac{\log\Big{(}\sum_{n=1}^{N}|a_{n}|\Big{)}}{\lambda_{N}},$$

$$\sigma_{u}(D)\leq\limsup_{N}\frac{\log\Big{(}\sup_{t\in\mathbb{R}}\big{|}\sum^{N}_{n=1}a_{n}e^{-\lambda_{n}it}\big{|}\Big{)}}{\lambda_{N}},$$

$${\sigma }_b(D)=\inf \sigma ,$$

$${\sigma }_b^{ext}(D)=\inf \sigma ,$$

$${\sigma }_b^{ext}(D)={\sigma }_u(D)$$

$${\mathcal{D}}_{\infty }^{ext}(\lambda )$$

$$\exists \text{leavevmode}l=l(\lambda )>0\text{leavevmode}\forall \text{leavevmode}\delta >0\text{leavevmode}\exists \text{leavevmode}C>0\text{leavevmode}\forall n\in \mathbb{N}:\text{leavevmode}\text{leavevmode}{\lambda }_{n+1}-{\lambda }_n\ge C{e}^{-(l+\delta ){\lambda }_n};$$

$$\forall \text{leavevmode}\delta >0\text{leavevmode}\exists \text{leavevmode}C>0\text{leavevmode}\forall \text{leavevmode}n\in \mathbb{N}:\text{leavevmode}\text{leavevmode}{\lambda }_{n+1}-{\lambda }_n\ge C{e}^{-e^{\delta {\lambda }_n}}.$$

$$\exists \text{leavevmode}x>0\text{leavevmode}\exists \text{leavevmode}C>0\text{leavevmode}\forall \text{leavevmode}n\in \mathbb{N}:\text{leavevmode}\text{leavevmode}{\lambda }_{n+1}-{\lambda }_n\ge C{e}^{-e^{x{\lambda }_n}}.$$

$$\sup_{[Re>0]}\bigg{|}\sum_{n=1}^{N}a_{n}n^{-s}\bigg{|}\leq C\log(N)\|D\|_{\infty}.$$

$$S_N:{\mathcal{D}}_{\infty }^{ext}(\lambda )\to {\mathcal{D}}_{\infty }(\lambda ),\text{leavevmode}\text{leavevmode}D\mapsto \sum _{n=1}^N{a}_n(D)e^{-{\lambda }_{n}s}.$$

Full text

††footnotetext: 2010 Mathematics Subject Classification: Primary††footnotetext: Key words and phrases: ††footnotetext:

1. Introduction

Within the last two decades the theory of ordinary Dirichlet series $\sum a_{n}n^{-s}$ saw a sort of renaissance. The study of these series in fact was one of the hot topics in mathematics at the beginning of the 20th. Among others, H. Bohr, Besicovitch, Bohnenblust, Hardy, Hille, Landau, Perron, M. Riesz, and Neder, were the leading mathematicians in this issue. The theory lived a sort of golden moment between the 1910’s and the 1930’s but after that it was somehow forgotten. Some 20 years ago the seminal article [33] of Hedenmalm, Lindqvist and Seip called again the attention from analysis to Dirichlet series. Since then a lot has been going on, and ordinary Dirichlet series have been studied with new techniques from functional and harmonic analysis.

Bohr’s main interest was to derive properties of Dirichlet series from the analytical properties of the holomorphic functions defined by them. It is well known that Dirichlet series converge on half planes, and where they converge, they define a holomorphic function. Bohr considered three abscissas for a given Dirichlet series $D=\sum a_{n}n^{-s}$: $\sigma_{c}(D)$, $\sigma_{u}(D)$, and $\sigma_{a}(D)$ that define the maximal half planes on which the series respectively converges, converges uniformly, or converges absolutely. He also considered a fourth abscissa, $\sigma_{b}(D)$, that gives the maximal half plane on which the series defines a bounded and holomorphic function. Then

[TABLE]

In [7] Bohr was interested on describing the absolute convergence abscissa of an ordinary Dirichlet series in terms of analytic properties of its limit function. He proved

[TABLE]

(see e.g. [26, Corollary 1.14]), and then considered the number

[TABLE]

that gives the maximal width of the band on which a Dirichlet series can converge uniformly but not absolutely. Bohr showed that $S\leq 1/2$. The problem of whether or not this was the correct value remained open for some 15 years, until Bohnenblust and Hille in [5] indeed proved that

[TABLE]

(see also [26, Theorem 4.1]). These ideas are the seeds of a recent revival of interest in the research area opened up by these early contributions. A new field emerged intertwining the classical work in novel ways with modern functional analysis, infinite dimensional holomorphy, probability theory as well as analytic number theory. As a consequence, a number of challenging research problems crystallized and were solved over the last decades. We refer to the monographs [36], [41], and [26] were many of the key elements of this new developments for ordinary Dirichlet series are described in detail.

A fundamental object in these investigations is given by the Banach space $\mathcal{D}_{\infty}$ of all ordinary Dirichlet series $D=\sum a_{n}n^{-s}$ which converge and define a bounded, and then necessarily holomorphic, function on $[Re>0]$ (endowed with the supremum norm on $[Re>0]$), and a celebrated result from [33] shows that $\mathcal{D}_{\infty}$ in fact equals the Hardy space $H_{\infty}(\mathbb{T}^{\infty})$ on the infinitely dimensional torus. Let us explain this in more detail. The infinitely dimensional torus $\mathbb{T}^{\infty}$ is the infinite product of $\mathbb{T}=\{w\in\mathbb{C}\colon|w|=1\}$ which forms a natural compact abelian group on which the Haar measure is given by the normalized Lebesgue measure. The characters on $\mathbb{T}^{\infty}$, so the elements in the dual group, consists of all monomials $z\mapsto z^{\alpha}$, where $\alpha=(\alpha_{k})\in\mathbb{Z}^{(\mathbb{N})}$ (all a finite sequence of integers), and $H_{\infty}(\mathbb{T}^{\infty})$ denotes the closed subspace of all $f\in L_{\infty}(\mathbb{T}^{\infty})$ such that the Fourier coefficient

[TABLE]

whenever $\alpha<0$ (in the sense that some $\alpha_{k}<0$). Then, based on Bohr’s work, it is proven in [33] (see again [26, Corollary 5.3]) that there is a unique linear isometry

[TABLE]

which preserves Fourier- and Dirichlet coefficients in the sense that

[TABLE]

where $\alpha=(\alpha_{1},\ldots,\alpha_{N},0\ldots)\in\mathbb{N}_{0}^{(\mathbb{N})}$ and $\mathfrak{p}=2,3,5\ldots$ the sequence of primes. We refer to this result as the Bohr-Hedenmalm-Lindqvist-Seip theorem (see also [26, Corollary 5.3]). The crucial point of its proof is the fact that each natural number has a unique prime number decomposition, as well as Kronecker’s theorem on Diophantine approximation (see e.g. [26, Proposition 3.4]): The continuous group homomorphism

[TABLE]

has dense range, which in particular implies that for each $n$ and $\alpha$ with $n=\mathfrak{p}^{\alpha}$ the following diagram commutes:

[TABLE]

The original proof of (3) goes a detour through infinite dimensional holomorphy using a result of Cole and Gamelin from [28]. Denote by $H_{\infty}(B_{c_{0}})$ all holomorphic (Fréchet differentiable) functions $f$ on the open unit ball $B_{c_{0}}$ of $c_{0}$, which endowed with the sup norm forms a Banach space. Then Cole and Gamelin show that there is a unique isometric linear bijection

[TABLE]

which preserves Fourier and monomial coefficients in the sense that for every multiindex $\alpha$

[TABLE]

(see [26, Theorem 5.1]). The equality of Banach spaces from (3) shows that a Dirichlet series $D$ belongs to $\mathcal{D}_{\infty}$ if and only if there is a function $f\in H_{\infty}(\mathbb{T}^{\infty})$ such that the Dirichlet coefficients $(a_{n}(D))$ and the Fourier coefficients $(\hat{f}(\alpha))$ coincide in the sense of (4), if and only if there is a function $f\in H_{\infty}(B_{c_{0}})$ such that the Dirichlet coefficients $(a_{n}(D))$ and the monomial coefficients $(\frac{\partial^{\alpha}g(0)}{\alpha!})$ coincide in the sense of (7). This links the theory of ordinary Dirichlet which generate bounded, holomorphic functions on the positive half plane intimately with Fourier analysis on the group $\mathbb{T}^{\infty}$ as well as infinite dimensional holomorphy on the open unit ball of $c_{0}$.

More generally, Bayart in [3] developed an $H_{p}$-theory of Dirichlet series. Recall that the Hardy space $H_{p}(\mathbb{T}^{\infty})\,,\,1\leq p\leq\infty,$ is the closed subspace of all $f\in L_{p}(\mathbb{T}^{\infty})$ such that $\hat{f}(\alpha)=0$ only if $\alpha<0$. Then the Banach spaces $\mathcal{H}_{p}$ of ordinary Dirichlet series by definition is the isometric image of $H_{p}(\mathbb{T}^{\infty})$ under the identification from (4):

[TABLE]

and the Bohr-Hedenmalm-Lindqvist-Seip theorem from (6) reads

[TABLE]

Again the Banach spaces $\mathcal{H}_{p}$ can be reformulated in terms of holomorphic functions in infinitely many variables. Instead of looking at holomorphic functions on $B_{c_{0}}$, we look at holomorphic functions on $\ell_{2}\cap B_{c_{0}}$, understood as an open subset of $\ell_{2}$. Then

[TABLE]

is the Banach space of all holomorphic functions $g:\ell_{2}\cap B_{c_{0}}\rightarrow\mathbb{C}$ for which

[TABLE]

Then for $1\leq p<\infty$ there is a unique isometric identity

[TABLE]

identifying Fourier coefficients and monomial coefficients in the sense of (7) (a detailed proof is given in [26, Chapter 13]).

Let us now face general Dirichlet series – our main object of interest. Given a frequency $\lambda:=(\lambda_{n})$ (i.e. a non-negative sequence of real numbers tending to $+\infty$), a $\lambda$-Dirichlet series is a formal sum

[TABLE]

with complex Dirichlet coefficients $(a_{n})$ and a complex variable $s$. Denote by $\mathcal{D}(\lambda)$ the space of all formal $\lambda$-Dirichlet series.

We start recalling some basic facts. The natural domains of convergence of Dirichlet series are half spaces (see [32, §II.2, Theorem 1]), and as in the ordinary case the ’abscissas’

[TABLE]

rule the convergence theory of general Dirichlet series. Again general Dirichlet series $D$ define holomorphic functions on $[Re>\sigma_{c}(D)]$, which relies on the fact that they converge uniformly on all compact subsets of $[Re>\sigma_{c}(D)]$ (see [32, §II.2, Theorem 2]).

There are useful Bohr-Cahen formulas for the abscissas $\sigma_{c}$ and $\sigma_{a}$, that are, given $D=\sum a_{n}e^{-\lambda_{n}s}$,

[TABLE]

where in each case equality holds if the left hand side is non negative. See [32, §II.6 and §II.7] for a proof. The formula for $\sigma_{u}$ (and its proof) extends from the ordinary case in [26, §1.1, Proposition 1.6] canonically to arbitrary $\lambda$’s:

[TABLE]

where again equality holds if the left hand side is non negative.

We point out that making the jump from the ordinary case $\lambda=(\log n)$ to arbitrary frequencies reveals serious difficulties and that many of the above ideas fail for general Dirichlet series. Much of the ordinary theory relies on Bohr’s theorem from (1), the fact that for each ordinary Dirichlet series the abscissa of uniform convergence and boundedness coincide. This phenomenon fails for general Dirichlet series. Further due to the prime number theorem each natural number $n$ has its prime number decomposition $n=\mathfrak{p}^{\alpha}$ and so the frequency $(\log n)$ can be written as a linear combination of $(\log p_{j})$ with natural coefficients. This intimately links the theory of ordinary Dirichlet series with the theory of holomorphic functions on polydiscs, and in particular with the theory of polynomials $\sum c_{\alpha}z^{\alpha}$ in finitely many complex variables. One of several consequences is that $m$-homogeneous Dirichlet series $\sum a_{n}n^{-s}$, i.e. $a_{n}\neq 0$ only if $n$ has $m$ prim factors, are linked with $m$-homogeneous polynomials. This way powerful tools enter the game, as e.g. polynomial inequalities (like the Bohnenblust-Hille inequalities, hypercontractivity of convolution with the Poisson kernel, etc.), $m$-linear forms, or polarization. So modern Fourier analysis and infinite dimensional holomorphy enrich the theory of ordinary Dirichlet series considerably. All this is carefully explained in [26].

Unfortunately facing general Dirichlet series many of these powerful bridges seem to collapse and new questions arise which make the theory of general Dirichlet quite challenging.

Nevertheless inspired by the ordinary theory we are able to introduce a Fourier analysis setting for the study of general Dirichlet series by restricting ourserlves to $\lambda$-Dirichlet series with Dirichlet coefficients that actually are Fourier coefficients defined by functions on compact abelian groups $G$ of a certain type; compact abelian groups allowing a continuous homomorphism $\beta\colon\mathbb{R}\to G$ with dense range.

2. Bohr’s theorem

To get started we define the space $\mathcal{D}_{\infty}(\lambda)$ analogously to the space $\mathcal{D}_{\infty}=\mathcal{D}_{\infty}((\log n))$.

Definition 2.1**.**

Let $\lambda$ be a frequency. Then $\mathcal{D}_{\infty}(\lambda)$ denotes the space of all $\lambda$-Dirichlet series $D$ which converge on $[Re>0]$, and define (a then necessarily holomorphic) bounded function there.

For instance, if $\lambda=(n)_{n=0}^{\infty}$, then looking at the transformation $z=e^{-s}$ we easily conclude that $\mathcal{D}_{\infty}((n))$ is simply $H_{\infty}(\mathbb{D})$, the space of all bounded and holomorphic functions on the open unit ball $\mathbb{D}$. And if $\lambda=(\log n)$, then we are in the ordinary case.

Recall that there is a unique coefficient preserving isometry identifying $H_{\infty}(\mathbb{D})$ and $H_{\infty}(\mathbb{T})$. Hence, in both cases, $\lambda=(n)$ and $\lambda=(\log n)$, the space $\mathcal{D}_{\infty}(\lambda)$ can be described in terms of Fourier analysis, that is, it can be considered as a Hardy space, namely $\mathcal{D}_{\infty}((n))=H_{\infty}(\mathbb{T})$ and $\mathcal{D}_{\infty}((\log n))=H_{\infty}(\mathbb{T}^{\infty})$. In view of these two examples the following question arises naturally: Given an arbitrary frequency $\lambda$, is it possible to describe $\mathcal{D}_{\infty}(\lambda)$ in terms of a sort of Hardy space on a compact abelian group?

The first step towards the solution of this problem, is to follow ideas of Bohr and Landau, who asked under which assumptions the abscissa $\sigma_{u}(D)$ can be described in terms of analytic properties of the limit function of $D$. Define

[TABLE]

where the infimum is taken over all $\sigma\in\mathbb{R}$ such that $D$ converges and defines a bounded function on $[\text{Re}>\sigma]$ (which then is automatically holomorphic). Additionally define, provided $\sigma_{c}(D)<\infty$,

[TABLE]

the infimum now taken over all $\sigma\in\mathbb{R}$ such that the limit function of $D$ allows a holomorphic and bounded extension to $[\text{Re}>\sigma]$. By definition $\sigma_{c}(D)\leq\sigma_{b}(D)\leq\sigma_{u}(D)\leq\sigma_{a}(D)$ and $\sigma_{b}^{ext}(D)\leq\sigma_{b}(D)$. But in general all these abscissas differ. For instance an example of Bohr from [11] shows that $\sigma_{c}(D)=\sigma_{b}^{ext}(D)=\sigma_{b}(D)=-\infty$ and $\sigma_{u}(D)=+\infty$ is possible.

A very prominent research project at the beginning of the 20th century was to find conditions for frequencies $\lambda$ under which

[TABLE]

holds for all somewhere convergent $\lambda$-Dirichlet series $D$. In this context the space

[TABLE]

of all somewhere convergent $\lambda$-Dirichlet series $D=\sum a_{n}e^{-\lambda_{n}s}$, which have a holomorphic and bounded extension to $[Re>0]$, appears naturally. Endow $\mathcal{D}_{\infty}^{ext}(\lambda)$ with the semi norm $\|D\|_{\infty}:=\sup_{[Re>0]}|f(s)|$, where $f$ is the (unique) extension of $D$. We will later in fact see that this always defines a norm – but that, in general, neither $\mathcal{D}_{\infty}(\lambda)$ nor $\mathcal{D}_{\infty}^{ext}(\lambda)$ form Banach spaces.

Much of the abstract theory of ordinary Dirichlet series is based on a fundamental theorem of Bohr from [8] which shows that every $D\in\mathcal{D}^{ext}_{\infty}((\log n))$ converges uniformly on all half spaces $[\text{Re}>\varepsilon]$ for all $\varepsilon>0$, and this then easily implies that $\mathcal{D}_{\infty}^{\text{ext}}((\log n))=\mathcal{D}_{\infty}((\log n))$ (see also [26, Theorem 1.13]). For certain classes of frequencies $\lambda$, Bohr [7] and Landau [38] extended this result to $\lambda$-Dirichlet series.

In [7] Bohr shows that $(\ref{problem})$ holds if $\lambda$ satisfies the following condition (we call it Bohr’s condition $(BC)$):

[TABLE]

roughly speaking this condition prevents the $\lambda_{n}$’s from getting too close too fast. Motivated by Bohr’s work we introduce the following definition.

Definition 2.2**.**

We say that a frequency $\lambda$ satisfies Bohr’s theorem, whenever every $D\in\mathcal{D}^{ext}_{\infty}(\lambda)$ converges uniformly on all half spaces $[\text{Re}>\varepsilon]\,,\,\,\varepsilon>0$, i.e. $\sigma_{b}^{ext}(D)=\sigma_{u}(D)$.

Observe that $\lambda=(\log n)$ satisfies $(BC)$ with $l=1$ and so Bohr’s theorem holds. In [38] Landau gave a weaker sufficient condition than $(BC)$ (we call it Landau’s condition $(LC)$), which extends the class of frequencies which satisfy Bohr’s theorem:

[TABLE]

We like to mention that in [39, §1] Neder went a step further and considered $\lambda$’s satisfying

[TABLE]

Then Neder proved that this condition is not sufficient for Bohr’s theorem by constructing, given some $x>0$, a Dirichlet series $D$ (belonging to some frequency $\lambda$) for which $\sigma_{c}(D)=\sigma_{a}(D)=x$ and $\sigma_{b}^{ext}(D)\leq 0$ holds, which in particular shows that $\mathcal{D}_{\infty}(\lambda)\subsetneq\mathcal{D}^{ext}_{\infty}(\lambda)$.

Analysing Bohr’s original proof from [7], Bohr’s theorem in the ordinary case was improved in [4] by a quantitative version (see again [26, Theorem 1.13]).

Theorem 2.3**.**

There is a constant $C>0$ such that for every $D\in\mathcal{D}_{\infty}((\log n))$ and all $N\geq 2$

[TABLE]

Note that in the ordinary case $\lambda=(\log n)$, this inequality together with $(\ref{sigmaU})$ implies $(\ref{problem})$.

Like in the work of Bohr, Landau only proves the qualitative version of the fact that each frequency $\lambda$ under his condition $(LC)$ satisfies Bohr’s theorem. To establish quantitative versions in the sense of (15) means to control the norm of the partial sum operator

[TABLE]

Using the summation method of typical means of order $k>0$ invented by M. Riesz (see Proposition 2.6), the following estimate of $\|S_{N}\|$, which holds under no further conditions on $\lambda$, is the main result of [43, Theorem 3.2].

Theorem 2.4**.**

Let $D=\sum a_{n}e^{-\lambda_{n}s}\in\mathcal{D}_{\infty}^{ext}(\lambda)$. Then for all $0<k\leq 1$ and $N\in\mathbb{N}$:

[TABLE]

where $C>0$ is a universal constant and $\Gamma$ denotes the gamma function.

As a consequence, assuming Bohr’s condition $(BC)$ for $\lambda$, the choice $k_{N}:=\frac{1}{\lambda_{N}}$, $N\geq 2$ (note that $\lambda_{1}=0$ is possible) leads to

[TABLE]

which reproves $(\ref{BTordinary})$ for $\lambda=(\log n)$. Assuming Landau’s condition $(LC)$, the estimate from Theorem 2.4 with $k_{N}:=e^{-\delta\lambda_{N}}$, $\delta>0$, gives

[TABLE]

the quantitative version of Bohr’s theorem under $(LC)$.

For particular frequencies $\lambda$ the bounds from (16) and (17) may be bad, which isn’t too surprising since these results in fact hold for all frequencies satisfying the conditions of Bohr or Landau. For instance, consider the case $\lambda=(n)$. Then the projection

[TABLE]

is nothing else than the convolution operator which assigns to every $f$ its convolution with the Dirichlet kernel $D_{N}$, and this immediately gives that $\|S_{N}\|=\|D_{N}\|_{1}\sim\log(N)$. To our knowledge the optimal upper and lower bounds for the norm of $S_{N}$ in the ordinary case $\lambda=(\log(n))$ are still unknown.

Problem 2.5**.**

Determine optimal bounds for $\|S_{N}\colon\mathcal{D}_{\infty}((\log n))\to\mathcal{D}_{\infty}((\log n))\|$.

The proof of Theorem 2.4 relies on the following independently interesting result from [43, Proposition 3.4] which was inspired by the work of Hardy and M. Riesz from [32].

Proposition 2.6**.**

Let $D=\sum a_{n}e^{-\lambda_{n}s}\in D_{\infty}^{ext}(\lambda)$ with extension $f$. Then for all $k>0$ the Dirichlet polynomials

[TABLE]

converge uniformly to $f$ on $[Re>\varepsilon]$ for all $\varepsilon>0$. Moreover,

[TABLE]

In the language of [32] Proposition 2.6 states that, given any order $k>0$, then on every halfplane $[Re>\varepsilon]$ the limit function of a Dirichlet series $D\in\mathcal{D}^{ext}_{\infty}(\lambda)$ is the uniform limit of its first typical means of order $k$.

Moreover, Proposition 2.6 gives a direct link to the theory of almost periodic functions on $\mathbb{R}$, and proves that $\mathcal{D}_{\infty}^{ext}(\lambda)$ in fact is a normed space (see Corollary 2.8). Note that a priori, $\|\cdot\|_{\infty}$ is only a semi norm, or equivalently, it is not clear whether $\mathcal{D}^{ext}_{\infty}(\lambda)$ can be considered as a subspace of $H_{\infty}[Re>0]$, the Banach space of all holomorphic and bounded functions on $[Re>0]$. Here it is important to distinguish Dirichlet series from their limit functions, and to prove that $\|\cdot\|_{\infty}$ in fact is a norm on $\mathcal{D}^{ext}_{\infty}(\lambda)$ requires to check that all Dirichlet coefficients of $D$ vanish provided $\|f\|_{\infty}=0$.

Recall that by definition a continuous function $f\colon\mathbb{R}\to\mathbb{C}$ is called (uniformly) almost periodic, whenever for every $\varepsilon>0$ there is a number $l>0$ such that for all intervals $I\subset\mathbb{R}$ with $|I|=l$ there is a translation number $\tau\in I$ such that $\sup_{x\in\mathbb{R}}|f(x-\tau)-f(x)|\leq\varepsilon$ (see [6] for more information). Then by a result of Bohr a bounded and continuous function $f$ is almost periodic if and only if it is the uniform limit of trigonometric polynomials on $\mathbb{R}$, which are of the form $p(t):=\sum_{n=1}^{N}a_{n}e^{-itx_{n}}$ for some $x_{1},\ldots,x_{N}\in\mathbb{R}$ (see e.g. [41, §1.5.2.2, Theorem 1.5.5]). In particular, the Dirichlet polynomials $R_{x}^{k}(D)$ stated in Proposition 2.6 considered as functions on vertical lines $[Re=\sigma]$ are almost periodic.

Corollary 2.7**.**

If $D\in\mathcal{D}_{\infty}^{ext}(\lambda)$ with extension $f$, then the function $f_{\sigma}(t):=f(\sigma+it)\colon\mathbb{R}\to\mathbb{C}$ is almost periodic and

[TABLE]

for all $\sigma>0$. In particular, $\sup_{n\in\mathbb{N}}|a_{n}|\leq\|f\|_{\infty}$.

This result is taken from [43, Corollary 3.8], and the next corollary is then an immediate consequence.

Corollary 2.8**.**

$\mathcal{D}_{\infty}^{ext}(\lambda)$, and consequently also its subspace $\mathcal{D}_{\infty}(\lambda)$, are normed spaces for any frequency $\lambda$.

Another particular consequence of Corollary 2.7 is as follows. Clearly, we may deduce from this corollary that $\|S_{N}\colon\mathcal{D}_{\infty}^{ext}(\lambda)\to\mathcal{D}_{\infty}(\lambda)\|\leq N$ for all $N$, so equality (12) follows, i.e. $\lambda$ satisfies Bohr’s theorem, whenever

[TABLE]

We like to mention that the number $L(\lambda)$ has the following geometric meaning in terms of abscissas. In [9, §3, Hilfssatz 3, Hilfssatz 2] Bohr proved that

[TABLE]

For instance we have that $L((n))=0$, and see as a consequence that for power series we up to $\varepsilon$ can’t distinguished between uniform convergence and boundedness of the limit function.

Besides $\lambda$’s with $(BC)$ or $(LC)$, there is another class of frequencies $\lambda$ for which Bohr’s theorem holds. In [12] Bohr proved that $\mathbb{Q}$-linearly independent frequencies $\lambda$ satisfy

[TABLE]

for all somewhere convergent $\lambda$-Dirichlet series $D$. The use of Kronecker’s theorem, which states that the set $\left\{(e^{-\lambda_{n}it})_{n}\mid t\in\mathbb{R}\right\}$ is dense in $\mathbb{T}^{\infty}$ whenever the real sequence $(\lambda_{n})$ is $\mathbb{Q}$-linearly independent, combined with Proposition 2.6 in [43, Theorem 4.7] lead to an alternative proof of this fact.

Theorem 2.9**.**

Let $D=\sum a_{n}e^{-\lambda_{n}s}\in\mathcal{D}^{ext}_{\infty}(\lambda)$ with a $\mathbb{Q}$-linearly independent frequency $\lambda$. Then $(a_{n})\in\ell_{1}$ and $\|(a_{n})\|_{1}=\|D\|_{\infty}$. Moreover,

[TABLE]

and $\lambda$ satisfies Bohr’s theorem. In particular,

[TABLE]

The following theorem summarizes some of the preceding results on frequencies $\lambda$ satisfying Bohr’s theorem.

Theorem 2.10**.**

A frequency $\lambda$ satisfies Bohr’s theorem, if one of following conditions holds:

• •

$L(\lambda)=0$**

• •

$\lambda$* is $\mathbb{Q}$-linearly independent*

• •

$\lambda$* satisfies $(LC)$ (or the stronger condition $(BC)$)*

Moreover, in each of these cases we have $\mathcal{D}_{\infty}^{\text{ext}}(\lambda)=\mathcal{D}_{\infty}(\lambda)$, but non of these conditions is necessary for Bohr’s theorem.

Unfortunately $\mathcal{D}_{\infty}(\lambda)$ may fail to be a Banach space. The following result from [43, Theorem 5.2] is inspired by a construction of Neder in [39].

Proposition 2.11**.**

Let $\lambda$ be a frequency. Then there is a strictly increasing sequence $(s_{n})$ of natural numbers such that $\mathcal{D}_{\infty}(\eta)$, where $\eta$ is the frequency obtained by ordering the set

[TABLE]

increasingly, is not complete and $\mathcal{D}_{\infty}(\eta)\subsetneq\mathcal{D}_{\infty}^{ext}(\eta)$. In particular, Bohr’s theorem fails for $\eta$.

The good news is that the completeness of $\mathcal{D}_{\infty}(\lambda)$ is guaranteed by several sufficient conditions on $\lambda$ which in concrete cases have a good chance to be checked. The following collection of results was proved in [43, Theorem 5.1] (for the case of $(BC)$ see also [19]).

Theorem 2.12**.**

$\mathcal{D}^{ext}_{\infty}(\lambda)$* is complete, if $L(\lambda)<\infty$. In particular, $\mathcal{D}_{\infty}(\lambda)$ is a Banach space and coincides with $\mathcal{D}_{\infty}^{ext}(\lambda)$ provided that $\lambda$ satisfies one of the following conditions:*

• •

$L(\lambda)=0$**

• •

$\lambda$* is $\mathbb{Q}$-linearly independent*

• •

$\lambda$* satisfies $(LC)$ and $L(\lambda)<\infty$ (this includes $(BC)$).*

Moreover, none of these conditions is necessary for the completeness of $\mathcal{D}_{\infty}(\lambda)$.

In view of the different nature of the stated conditions in Theorem 2.10 and Theorem 2.12, it seems that we are far away from a characterization of completeness of $\mathcal{D}_{\infty}(\lambda)$ (or $\mathcal{D}_{\infty}^{ext}(\lambda)$).

Problem 2.13**.**

Characterize completeness of $\mathcal{D}_{\infty}(\lambda)$ and $\mathcal{D}_{\infty}^{ext}(\lambda)$.

In this context more interesting questions appear naturally.

Problem 2.14**.**

Is there any relation between the $\lambda$’s satisfying Bohr’s theorem and the $\lambda$’s for which $\mathcal{D}_{\infty}(\lambda)$ is complete?

For instance the frequency $\lambda:=(\sqrt{\log n})$ fulfils $(LC)$ (and so satisfies Bohr’s theorem), but we don’t know wether $\mathcal{D}_{\infty}((\sqrt{\log n}))$ is complete.

Problem 2.15**.**

Is $\mathcal{D}_{\infty}((\sqrt{\log n}))$ complete?

Recall Theorem 2.10 where we conditions on frequencies under which Bohr’s theorem hold, that is $\sigma_{b}^{ext}(D)=\sigma_{b}(D)$ for all $\lambda$-Dirichlet series $D$.

Problem 2.16**.**

Find a reasonable condition on the frequency $\lambda$ which is weaker than $(LC)$, but is sufficient for the equality $\sigma_{b}^{ext}(D)=\sigma_{b}(D)$ for all $\lambda$-Dirichlet series $D$.

By Theorem 2.12, $\mathcal{D}_{\infty}(\lambda)$ is complete, if $\sigma_{b}^{ext}(D)=\sigma_{b}(D)$ for all $\lambda$-Dirichlet series $D$ and $L(\lambda)<\infty$. So any contribution to Problem 2.16 may give partial solutions to Problem 2.14.

We finish summarizing a few relations of the three conditions $(BC)$, $(LC)$ and $L(\lambda)<\infty$ (see [43, Remark 4.1]).

Remark 2.17**.**

---

• •

$(BC)$* implies $L(\lambda)<\infty$ and $(LC)$.*

• •

$(LC)$* plus $L(\lambda)<\infty$ does not necessarily imply $(BC)$.*

• •

$L(\lambda)<\infty$* does not necessarily imply $(LC)$, and so neither $(BC)$.*

• •

$(LC)$* does not necessarily imply $L(\lambda)<\infty$, and so neither $(BC)$.*

3. Hardy spaces of general Dirichlet series

Now we start our $\mathcal{H}_{p}$-theory on general Dirichlet series. As already mentioned in the introduction we restrict ourselves to general Dirichlet series with Dirichlet coefficients which actually are Fourier coefficients of functions on certain compact abelian groups. This has several advantages. One is that the class of all general Dirichlet series simply is too large to obtain a good understanding. Assuming that the Dirchlet coefficients are Fourier coefficients gives more structure and allows to use tools from harmonic analysis like the Hausdorff-Young inequality or Plancherel’s theorem (among others). A further advantage of our setting is that Bayart’s $\mathcal{H}_{p}$-theory of ordinary Dirichlet series embeds in a natural way. Whereas the $\mathcal{H}_{p}$-theory of ordinary Dirichlet series is basically Fourier analysis on the infinite dimensional torus $\mathbb{T}^{\infty}$, this group fails to be the right model for general Dirichlet series. In fact, the Bohr compactification $\overline{\mathbb{R}}$ of $\mathbb{R}$ and products of $\widehat{\mathbb{Q}_{d}}$ (the dual group of the rationals endowed with the discrete topology) turn out to be suitable substitutes. Finally, fixing some $\lambda$, regarding the different realisations of $\lambda$-Dirichlet series of this type, another feature of our approach is that the $\mathcal{H}_{p}$-theory of general Dirichlet series we intend to present will be independent of the chosen suitable group for $\lambda$.

We are interested in the subclass of all compact abelian groups $G$ which allow a continuous homomorphism $\beta\colon\mathbb{R}\to G$ with dense range. We call such pairs $(G,\beta)$ Dirichlet groups. Hence, given such a pair, the characters $x=e^{-ix\cdot}\in\widehat{\beta}(\widehat{G})\subset\widehat{\mathbb{R}}$ are precisely those for which there is a unique character $h_{x}\in\widehat{G}$ with $e^{-ix\cdot}=h_{x}\circ\beta$; to understand this recall that $\mathbb{R}=\hat{\mathbb{R}},x\to e^{-ix\cdot}$ is a group isomorphism. In particular, we have

[TABLE]

Then the following notion (first given in [24]) turns out to be fundamental for our purposes.

Definition 3.1**.**

Let $\lambda$ be a frequency and $(G,\beta)$ a Dirichlet group. Then $G$ is called a $\lambda$-Dirichlet group whenever $\lambda\subset\widehat{\beta}(\widehat{G})$, which means that the following diagram commutes for every $n\in\mathbb{N}$:

[TABLE]

Let us give examples. Denoting by $\mathfrak{p}:=(p_{n})$ the sequence of prime numbers, the compact group $\mathbb{T}^{\infty}$ together with the mapping

[TABLE]

forms a $(\log n)$-Dirichlet group (see again Kronecker’s theorem). This example keeps us in track to recover results on ordinary Dirichlet series. The ’mother’ of all possible examples is as follows: Given a subgroup $U$ of $\mathbb{R}$, the topological group $\widehat{(U,d)}$ together with the mapping

[TABLE]

forms a Dirichlet group. So in particular, for $U=\mathbb{Z}$ and identifying $\mathbb{T}=\widehat{\mathbb{Z}}$, we obtain the $(n)$-Dirichlet group $(\mathbb{T},\beta_{\mathbb{T}})$, where

[TABLE]

The compact abelian group $\overline{\mathbb{R}}:=\widehat{(\mathbb{R},d)}$ is the so-called Bohr compactification of $\mathbb{R}$ which forms a $\lambda$-Dirichlet group for all frequencies $\lambda$ with the embedding

[TABLE]

Besides $\overline{\mathbb{R}}$, countable products of $\widehat{\mathbb{Q}_{d}}:=\widehat{\left(\mathbb{Q},d\right)}$, where $d$ denotes the discrete topology, also form ’universal’ Dirichlet groups. To explain this , let $B:=(b_{1},b_{2},\ldots)$ be a $\mathbb{Q}$-linearly independent sequence of real numbers of length $N\in\mathbb{N}\cup\{\infty\}$. Then

[TABLE]

is an injective homomorphism, and hence its dual map

[TABLE]

has dense range. Since $\prod_{n=1}^{N}\widehat{\mathbb{Q}_{d}}=\widehat{\bigoplus_{n=1}^{\infty}\mathbb{Q}_{d}}$, the pair $\left(\prod_{n=1}^{N}\widehat{\mathbb{Q}_{d}},\widehat{T_{B}}\right)$ is a Dirichlet group, which is ’universal’ in the following sense: For every frequency $\lambda$ there is a suitable $B$ such that $\left(\prod_{n=1}^{N}\widehat{\mathbb{Q}_{d}},\widehat{T_{B}}\right)$ is a $\lambda$-Dirichet group.

To see this, let us recall that to every $\lambda$ there is a $\mathbb{Q}$-linearly independent sequence $B:=(b_{n})$ of real numbers, called basis for $\lambda$ (which can always be chosen as a subsequence of $\lambda$), such that $\lambda_{n}=\sum r^{n}_{k}b_{n}$ for some (unique) finite rational sequence $(r^{n}_{k})$. In this case, $R:=(r^{n}_{k})_{n,k}$ is said to be a Bohr matrix of $\lambda$ with respect to the basis $B$ and we write $\lambda=(R,B)$. Hence $\left(\prod_{n=1}^{N}\widehat{\mathbb{Q}_{d}},\widehat{T_{B}}\right)$ is a $\lambda$-Dirichlet group for every $\lambda$ with decomposition $\lambda=(R,B)$.

Given a $\lambda$-Dirichlet group $(G,\beta)$ and $1\leq p\leq\infty$, we define the Banach space

[TABLE]

and use it to define the following natural scale of Hardy spaces of general Dirichlet series.

Definition 3.2**.**

The Hardy space $\mathcal{H}_{p}(\lambda)$ of $\lambda$-Dirichlet series is the space of all $\sum a_{n}e^{-\lambda_{n}s}$ for which there is some $f\in H_{p}^{\lambda}(G)$ such that $a_{n}=\widehat{f}(h_{\lambda_{n}})$ for all $n$.

Together with the norm $\|D\|_{p}:=\|f\|_{p}$ the space $\mathcal{H}_{p}(\lambda)$ clearly forms a Banach space, and then by definition the Bohr map

[TABLE]

gives an isometric onto isomorphism

[TABLE]

The following fact, proved in [24, Theorem 3.19], is fundamental.

Theorem 3.3**.**

$\mathcal{H}_{p}(\lambda)$* is independent of the chosen $\lambda$-Dirichlet group $G$.*

So the above definition of $\mathcal{H}_{p}(\lambda)$ actually coincides with Bayart’s definition of $\mathcal{H}_{p}$ from [3] in the ordinary case (see also [26, §11]). Moreover, recall that in this case the groups $\mathbb{T}^{\infty}$ and $\overline{\mathbb{R}}$ are suitable $(\log n)$-Dirichlet groups, which immediately leads to the following consequence extending (3).

Corollary 3.4**.**

For all $1\leq p\leq\infty$ we have

[TABLE]

where $\widehat{f}(h_{\log n})=a_{n}(D)=\widehat{g}(\alpha)$, if $n=\mathfrak{p}^{\alpha}.$

Let us mention that there also is an internal description of $\mathcal{H}_{p}(\lambda)$ through $\lambda$-Dirichlet polynomials without considering $\lambda$-Dirichlet groups. We denote by $Pol(\lambda)$ the space of all $\lambda$-Dirichlet polynomials $D(s)=\sum_{n=1}^{N}a_{n}e^{-\lambda_{n}s}$. For such polynomials we define

[TABLE]

Then this limit exists and gives a norm on $Pol(\lambda)$ (see e.g. [41, Theorem 1.5.6] or [26, Theorem 11.9]). The following description was noted in [24, Theorem 3.25].

Theorem 3.5**.**

Let $1\leq p<\infty$ and $\lambda$ be a frequency. Then the space $\mathcal{H}_{p}(\lambda)$ is the completion of $(Pol(\lambda),\|\cdot\|_{p})$.

3.1. Frequencies of integer type

Recall that given a frequency $\lambda$ there is a basis $B$ for $\lambda$ and a Bohr matrix $R$ such that $\lambda=(R,B)$. Let us look at the ordinary case $\lambda=(\log n)$. Then $\log n=\sum\alpha_{j}\log p_{j}$ whenever $n=\mathfrak{p}^{\alpha}$, and hence a basis is given by $B=(\log p_{j})$ and every multi index $\alpha>0$ appears as a row in the corresponding Bohr matrix $R$. Moreover recall that $\mathbb{T}^{\infty}$ is a $(\log n)$-Dirichlet group with $\beta:\mathbb{R}\to\mathbb{T}^{\infty},\,\beta(t):=\mathfrak{p}^{-it}$.

More generally, we call of frequency $\lambda$ of integer (natural) type, if there is a basis $B=(b_{n})$ such that the Bohr matrix $R$ associated to $B$ and $\lambda$ only has integer (natural) entries. In this case $\mathbb{T}^{\infty}$ is a $\lambda$-Dirichlet group with $\beta(t):=e^{-itB}:=(e^{-itb_{1}},e^{-itb_{2}},\ldots)$, $t\in\mathbb{R}$, and the rows of the corresponding Bohr matrix are multi indices $\alpha$ (with integer entries). Actually one can show that a frequency $\lambda$ is of integer type if and only if there is a homomorphism $\beta\colon\mathbb{R}\to\mathbb{T}^{\infty}$ such that $(\mathbb{T}^{\infty},\beta)$ is a $\lambda$-Dirichlet group (see [24, Remark 3.32]).

To see an example, consider the set $M:=\{n+\sqrt{2}m\mid n,m\in\mathbb{Z}\}$ which is dense in $\mathbb{R}$. Then any frequency $\lambda\subset M$ is of integer type with basis $B=(1,\sqrt{2})$. See [24, Example 3.36] for a frequency which is not of integer type.

Let $B$ be a basis of length $N\in\mathbb{N}\cup\{\infty\}$ for some frequency $\lambda$ of integer type with Bohr matrix $R$, and let us write $\alpha\in R$ if the multi index $\alpha$ appears as a row in $R$. Then we define $H_{p}^{R}(\mathbb{T}^{N})$ to be the space of all $g\in L_{p}(\mathbb{T}^{N})$ for which $\widehat{g}(\alpha)\neq 0$ implies $\alpha\in R$. By Theorem 3.3, the $\mathcal{H}_{p}(\lambda)$’s can be identified with $H_{p}^{R}(\mathbb{T}^{N})$ in the following sense (see Theorem [24, Theorem 3.30]).

Theorem 3.6**.**

Let $1\leq p\leq\infty$ and $\lambda=(R,B)$ a frequency of integer type with basis of length $N\in\mathbb{N}\cup\{\infty\}$. Then there is a unique onto isometry

[TABLE]

such that $\widehat{g}(\alpha)=a_{n}(D)$ for all multi indices $\alpha\in R$ and $\lambda_{n}=\sum\alpha_{j}b_{j}$.

An immediate consequence is that the ordinary $\mathcal{H}_{p}$’s in the following sense are the largest spaces for $\lambda$’s of natural type.

Corollary 3.7**.**

Let $1\leq p\leq\infty$ and $\lambda=(R,B)$ a frequency of natural type. Then there is a unique into isometry

[TABLE]

such that $a_{n}=b_{\mathfrak{p}^{\alpha}}$ for all $\alpha\in R$, where $\lambda_{n}=\sum\alpha_{j}b_{j}$.

4. Some structure theory

We summarize further properties of $\mathcal{H}_{p}(\lambda)$ which extend important key stones from Bayart’s theory of ordinary Dirichlet series to our new theory of general Dirichlet series.

4.1. Coincidence of $\boldsymbol{H_{\infty}}$’s

Note first that there are now two ‘$H_{\infty}$-spaces of $\lambda$-Dirichlet series’ around, namely $\mathcal{D}_{\infty}(\lambda)$ and $\mathcal{H}_{\infty}(\lambda)$. Recall that by Proposition 2.11 there are frequencies $\lambda$ such that $\mathcal{D}_{\infty}(\lambda)$ is not complete, hence in these cases $\mathcal{D}_{\infty}(\lambda)\neq\mathcal{H}_{\infty}(\lambda)$. The following result is given in [24, Theorem 4.10, 4.12], and an far reaching extension of the Bohr-Hedenmalm-Lindqvist-Seip theorem from (8).

Theorem 4.1**.**

For every frequency $\lambda$ there is a coefficient preserving injective contraction

[TABLE]

and if $\mathcal{D}^{ext}_{\infty}(\lambda)=\mathcal{D}_{\infty}(\lambda)$ and $L(\lambda)<\infty$ (see again Theorem 2.10), then the embedding is even an isometric equality

[TABLE]

We mention an interesting by-product of Theorem 4.1 which is an immediate consequence of the definition of $\mathcal{H}_{2}(\lambda)$ and Parseval’s equality.

Corollary 4.2**.**

For each $D\in\mathcal{D}^{ext}_{\infty}(\lambda)$ we have $(a_{n}(D))\in\ell_{2}$ with $\|(a_{n}(D))\|_{2}\leq\|D\|_{\infty}$, i.e. the embedding

[TABLE]

is a well-defined contraction.

4.2. Schauder bases

Given a frequency $\lambda$ and $1<p<\infty$, the following question is fundamental: Do the $e^{-\lambda_{n}s}$ form a Schauder basis of $\mathcal{H}_{p}(\lambda)$? In the ordinary case $\lambda=(\log n)$ the answer is affirmative as discovered in [1], and we will see that the same result holds true whenever $\lambda$ is an arbitrary frequency.

In (21) we note that, given a Dirichlet group $(G,\beta)$, we have

[TABLE]

This in particular shows that the dual group $\widehat{G}$ inherits the order of $\mathbb{R}$, and hence we deduce from [42, Theorem 8.7.2.], that the ’Riesz projection’

[TABLE]

is bounded on the subspace $Pol(G)$ of all polynomials in $\mathcal{H}_{p}(G)$, $1<p<\infty$. Then standard arguments shows the following important theorem from [24, Theorem 4.16].

Theorem 4.3**.**

Let $1<p<\infty$ and $\lambda$ be a frequency. Then the monomials $e^{-\lambda_{n}s}$ form a Schauder basis for $\mathcal{H}_{p}(\lambda)$.

An equivalent formulation of Theorem 4.3 is that for $1<p<\infty$ all projections

[TABLE]

are uniformly bounded. But for the border cases $p=1$ and $p=\infty$ this in general is false (e.g., for the frequencies $\lambda=(\log n)$ or $\lambda=(n)$). Upper bounds for the growth of the partial sum operators in $\mathcal{H}_{\infty}(\lambda)$ were given in (16) and (17). The following result handles the case $p=1$, and its proof in a sense reduces to the case $p=\infty$ (see [24, Proposition 4.17] and [26, 12.5] in the ordinary case).

Proposition 4.4**.**

Let $\lambda$ be a frequency. Assuming $(BC)$ for $\lambda$ there is a constant $C=C(\lambda)$ such that for all $N\geq 2$

[TABLE]

and, assuming $(LC)$ and $L(\lambda)<\infty$, for every $\delta$ there is a constant $D=D(\delta,\lambda)$ such that for all $N$

[TABLE]

4.3. Brothers Riesz theorem

Recall the classical brothers Riesz theorem with states that the Hardy space $\mathcal{H}_{1}((n))=H_{1}(\mathbb{T})$ coincides with the space $M(\mathbb{T})$ of all bounded, regular and analytic Borel measures on $\mathbb{T}$. The corresponding result in ordinary case reads $\mathcal{H}_{1}((\log n))=H_{1}(\mathbb{T}^{\infty})=M(\mathbb{T}^{\infty})$ (due to Helson and Lowdenslager from [37]). See also [26, Theorem 13.5] for a proof within the setting of ordinary Dirichlet series. The brothers Riesz theorem extends to the case of general Dirichlet series.

Theorem 4.5**.**

Let $\lambda$ be any frequency and let $G$ a $\lambda$-Dirichlet group. Then the map

[TABLE]

is an onto isometry. In particular $\mathcal{H}_{1}(\lambda)=M_{\lambda}(G)$.

Actually it was discovered in [24, Theorem 4.25] that this is a fairly simple consequence of a more general result due to [29, Theorem 4]; see [24, §4.7] for a discussion on this.

4.4. Montel theorem

In [3, Lemma 18] Bayart proved that for every bounded sequence $(D^{N})\subset\mathcal{D}_{\infty}((\log n))$ there is a subsequence $(D^{N_{k}})$ and $D\in\mathcal{D}_{\infty}((\log n))$ such that $(D^{N_{k}})$ converges to $D$ on $[Re>\varepsilon]$ for all $\varepsilon>0$. This fact, sometimes called ’Montel’s theorem’, extends to the following classes of $\lambda$’s. See also [41], and [26, Theorem 3.11], where Bayart’s Montel theorem is deduced from a Montel type theorem for $H_{\infty}(B_{c_{0}})$ combined with (6) and (8).

Theorem 4.6**.**

Let $\lambda$ satisfy $L(\lambda)=0$, or $L(\lambda)<\infty$ and $(LC)$, or let $\lambda$ be $\mathbb{Q}$-linearly independent. Then for every $1\leq p\leq\infty$ and every bounded sequence $(D^{N})\subset\mathcal{H}_{p}(\lambda)$ there is a subsequence $(D^{N_{k}})$ and $D\in\mathcal{H}_{p}(\lambda)$ such that for all $\varepsilon>0$ the translations about $\varepsilon$

[TABLE]

converge to $D$ in $\mathcal{H}_{p}(\lambda)$ as $k$ tends to $\infty$.

The proof in a sense reduces the $\mathcal{H}_{p}(\lambda)$-case to the $\mathcal{H}_{\infty}(\lambda)$-case. This reduction modifies an idea from [27] showing that, under the assumption of the completeness of $\mathcal{D}_{\infty}(\lambda)$, the map

[TABLE]

is an into isometry; here $\mathcal{D}_{\infty}(\lambda,\mathcal{H}_{p}(\lambda))$ stands for the Banach space of all $\lambda$-Dirichlet series $\sum A_{n}e^{-\lambda_{n}s}$ with coefficients $A_{n}\in\mathcal{H}_{p}(\lambda)$, which converge and are bounded on $[\text{Re}>0]$.

4.5. Hilbert’s criterion

As already mentioned the ’ordinary’ $\mathcal{D}_{\infty}$ isometrically equals $H_{\infty}(B_{c_{0}})$, identifying Dirichlet and monomial coefficients. A crucial argument in the proof of this result (see [26, §2.3]) is that a continuous function $f\colon B_{c_{0}}\to\mathbb{C}$ is in $H_{\infty}(B_{c_{0}})$ if and only if all restriction maps $f_{N}\colon\mathbb{D}^{N}\to\mathbb{C}$ are in $H_{\infty}(\mathbb{D}^{N})$ and $\sup_{N}\|f_{N}\|_{\infty}<\infty$.

Formulated for ordinary Dirichlet series this shows that $D=\sum a_{n}n^{-s}\in\mathcal{D}_{\infty}$ if and only if all of its so-called $N$-th abschnitte $D|_{N}=\sum a_{n}n^{-s}\,,$ where the sum is only taken over those $n$ which have only the first $N$ primes as divisors, belong to $\mathcal{D}_{\infty}$ with uniformly bounded norms. In more vague terms, $D\in\mathcal{D}_{\infty}$ if and only if all its finite dimensional blocks are in $\mathcal{D}_{\infty}$ with uniformly bounded norms (see also [26, Theorem 3.11]).

This phenomenon is also true in the general case. Given some decomposition $\lambda=(R,B)$ the $N$-th abschnitt $D|_{N}$ of a $\lambda$-Dirichlet series $D$ is the sum $\sum a_{n}(D)e^{-\lambda_{n}s}$, where $a_{n}(D)\neq 0$ implies that $\lambda_{n}$ depends only on the first $N$ basis elements $b_{1},\ldots,b_{N}$.

As proven in [24, Theorem 4.22], general Dirichlet series $D\in\mathcal{H}_{p}(\lambda)$ under appropriate assumptions on the frequency are again ’determined by their finite dimensional parts’.

Theorem 4.7**.**

Let $1\leq p\leq\infty$ and let $\lambda=(R,B)$ be a frequency satisfying one of the conditions of Theorem 4.6. Let $D$ be a formal $\lambda$-Dirichlet series. Then $D\in\mathcal{H}_{p}(\lambda)$ if and only if the $N$-th abschnitt $D|_{N}\in\mathcal{H}_{p}(\lambda)$ for all $N\in\mathbb{N}$ and $\sup_{N}\|D|_{N}\|_{p}<\infty$. Moreover, in this case $\|D\|_{p}=\sup_{N\in\mathbb{N}}\|D|_{N}\|_{p}.$

By Theorem 4.1 we obtain the following particular case: If $\lambda$ satisfies $\sigma_{b}^{ext}=\sigma_{b}$ and $L(\lambda)<\infty$, then $D\in\mathcal{D}_{\infty}(\lambda)$ if and only if $D|_{N}\in\mathcal{D}_{\infty}(\lambda)$ for all $N\in\mathbb{N}$ and $\sup_{N}\|D|_{N}\|_{\infty}<\infty$.

4.6. Helson’s theorem

A celebrated result of Helson [35] on general Dirichlet series $\sum a_{n}e^{-\lambda_{n}s}$ states that if $\lambda$ satisfies $(BC)$ and $(a_{n})$ is $2$-summable, then for almost all homomorphism $\omega\colon(\mathbb{R},+)\to\mathbb{T}$ the Dirichlet series $\sum a_{n}\omega(\lambda_{n})e^{-\lambda_{n}s}$ converges on the open right half plane $[Re>0]$, or equivalently, for all $u>0$ the series $\sum a_{n}e^{-\lambda_{n}u}h_{\lambda_{n}}$ converges almost everywhere on $\overline{\mathbb{R}}$ (note that if we for all rational $u$’s collect zero sets in $\overline{\mathbb{R}}$, then we get the aforementioned result). See also [33, Theorem 4.4] for the ordinary case.

Using our teminology, let us rephrase this result for square summable functions on Dirichlet groups: Given a frequency $\lambda$ with $(BC)$, a $\lambda$-Dirichlet group $(G,\beta)$, and some $f\in H_{2}^{\lambda}(G)$, then for every $u>0$ the series

[TABLE]

converges almost everywhere in $G$.

This result extends to functions in $H_{p}^{\lambda}(G)$. To see this recall that for $u>0$ the Poisson kernel

[TABLE]

is integrable with $\|P_{u}\|_{1}=1$. Hence, by the Riesz representation theorem, for every $\lambda$-Dirichlet group $(G,\beta)$ the functional

[TABLE]

defines a measure on $G$ which we call the ’Poisson measure’ on $G$ denoted by $p_{u}$.

The following theorem from [25] is a far reaching extension of Helson’s theorem. It in fact shows that, given an appropriate frequency $\lambda$, Helson’s theorem extends to functions $f$ from our scale $H_{p}^{\lambda}(G),1\leq p<\infty$, modelled on $\lambda$-Dirichlet groups $G$. Moreover, it describes the $G$-a.e. pointwise limit of the Fourier series $\sum_{n=1}^{\infty}\widehat{f}(h_{\lambda_{n}})e^{-\lambda_{n}u}h_{\lambda_{n}}$ in terms of convolution of $f$ with the Poisson measure $p_{u}$ on $G$, and – most important – it adds the relevant maximal inequality.

Theorem 4.8**.**

Let $(G,\beta)$ be a $\lambda$-Dirichlet group for a frequency with $(BC)$, and $f\in H_{p}^{\lambda}(G)$ with $1\leq p<\infty$. Then for every $u>0$

[TABLE]

almost everywhere on $G$. Moreover, there is a constant $C=C(u,\lambda)$ such that for all $f\in H_{p}^{\lambda}(G)$

[TABLE]

4.7. Vertical Limits

Let us reformulate Theorem 4.8 in terms of general Dirichlet series. Given a $\lambda$-Dirichlet series $D(s)=\sum a_{n}e^{-\lambda_{n}s}$ and a $\lambda$-Dirichlet group $G$, we for $\omega\in G$ call the Dirichlet series

[TABLE]

a vertical limit of $D$. The following characterization of vertical limits justifies their name ([24, Proposition 4.6]).

Remark 4.9**.**

Let $(G,\beta)$ be a $\lambda$-Dirichlet group, and $D=\sum a_{n}e^{-\lambda_{n}s}$ a $\lambda$-Dirichlet series with $\sigma_{a}(D)\leq 0$. Then

• (1)

For every $\omega\in G$ there is a sequence $(\tau_{k})\subset\mathbb{R}$ such for all $\varepsilon>0$ the ’vertical translations’

[TABLE]

converge to $D^{\omega}$ uniformly on $[Re>\varepsilon]$.

• (2)

Assume conversely that the $(D_{i\tau_{k}})$ for some $(\tau_{k})\subset\mathbb{R}$ converge uniformly on $[Re>\varepsilon]$ to a holomorphic function $f$ for every $\varepsilon>0$. Then there is $\omega\in G$ such that $f(s)=D^{\omega}(s)$ for all $s\in[Re>0]$ .

Note that the notion of vertical limits allows to rephrase Helson’s theorem from the preceding section as follows: Given the $\lambda$-Dirichlet group $G=\overline{\mathbb{R}}$ and $D\in\mathcal{H}_{2}(\lambda)$, then $\overline{\mathbb{R}}$-almost all vertical limits $D^{\omega}$ converge on $[Re>0]$.

The following result is then a straight forward reformulation of Theorem 4.8.

Theorem 4.10**.**

Let $(G,\beta)$ be a $\lambda$-Dirichlet group for a frequency with $(BC)$, and $D\in\mathcal{H}_{p}(\lambda)$ with $1\leq p<\infty$. Then almost all vertical limits

[TABLE]

converge on $[Re>0]$. Moreover, for all $u>0$ there is a constant $C=C(u,\lambda)$ such that for all $D\in\mathcal{H}_{p}(\lambda)$

[TABLE]

Remark 4.11**.**

Note that in the maximal inequality $(\ref{ineqmaxiDirichlet})$ the constant $C=C(u,\lambda$) is independent of $p$. Hence, if $p\to\infty$, then we recover that frequencies with $(BC)$ satisfy Bohr’s theorem, i.e. for every somewhere convergent $\lambda$-Dirichlet series $D$ we have $\sigma_{b}^{ext}(D)=\sigma_{u}(D)$.

Again convolution allows to relate the values of the vertical limits $D^{\omega}$ of a Dirichlet series $D\in\mathcal{H}_{p}(\lambda)$ and its associated function $f\in H_{p}^{\lambda}(G)$. The vertical limits $D^{\omega}$ in some sense recover the associated function $f$ on vertical lines. Note that, given $f\in L_{p}(G)$, almost all $\omega\in G$ define a measurable function $f_{\omega}(t):=f(\omega\beta(t))$, which itself is defined for almost all $t\in\mathbb{R}$. Hence, the convolution

[TABLE]

is defined almost everywhere on $\mathbb{R}$.

Corollary 4.12**.**

Let $1\leq p<\infty$, $\lambda$ satisfy $(BC)$, and $(G,\beta)$ be a $\lambda$-Dirichlet group. If $f\in H_{p}^{\lambda}(G)$ and $D\in\mathcal{H}_{p}(\lambda)$ are associated to each other, then for all $u>0$ and almost all $t\in\mathbb{R}$

[TABLE]

Let us consider the particular case of ordinary Dirichlet series. We denote by $\Xi$ the set of all completely multiplicative characters $\chi\colon\mathbb{N}\to\mathbb{T}$ (that is $\chi(nm)=\chi(n)\chi(m)$ for all $m$,$n$) which with the pointwise multiplication forms an abelian group. Looking at the group isomorphism

[TABLE]

where $\mathfrak{p}=(p_{n})$ again denotes the sequence of prime numbers, we see that $\Xi$ also forms a compact abelian group, and its Haar measure $d\chi$ is the push forward measure of $dz$ through $\iota^{-1}$. Then applying Theorem 4.8 to the $(\log n)$-Dirichlet group $\Xi$ we obtain the following consequence.

Corollary 4.13**.**

Let $1\leq p<\infty$. Then for all $u>0$ there is a constant $C=C(u)$ such that for all Dirichlet series $\sum a_{n}n^{-s}\in\mathcal{H}_{p}(\lambda)$

[TABLE]

In particular, almost all vertical limits $D^{\chi}(s)=\sum a_{n}\chi(n)n^{-s}$, $\chi\in\Xi$, converge on $[Re>0]$.

Remark 4.14**.**

Helson in [36] deduces from this result the curious fact that Riemann’s conjecture holds for almost all randomized $\zeta$-functions $\sum_{n}\chi(n)n^{-s}$, in the sense that almost all of them converge pointwise on $[\text{Re}>1/2]$ without having any zero in this half plane (see also [33]).

Bayart in [3] deduces the ’almost everywhere part’ of Corollary 4.13 for the case $p=2$ from the Menchoff-Rademacher theorem on almost everywhere convergence of orthonormal series, and then he extends it to the range $1\leq p<\infty$ using the ’hypercontractivity of the Poisson kernel’.

Let us explain what is meant by this. Recall the definition of the Poisson kernel on $\mathbb{T}$

[TABLE]

and consider for all $1\leq p<q<\infty$ the following convolution operator

[TABLE]

An easy computation shows that $\|P_{r}\|\leq\frac{1}{1-r}$ (independent of $p,q$), but this bound improves considerably depending on the relation of $p,q,r$. Weissler in [45] shows that $\|\mathcal{P}_{r}\|=1$ if and only if $r^{2}\leq\frac{p}{q}$; a result which is refered to as the ’hypercontractivity of the Poisson kernel’. Then using this result ’in each variable separately’ shows that for all $u>0$ and $1\leq p,q<\infty$ the translation operator

[TABLE]

is well-defined and bounded (see [26, Theorem 12.9] for the details on all that). This explains, why in the ordinary case ’the almost everywhere part’ of Corollary 4.13 follows in the range $1\leq p<\infty$, once this result is proven for $p=2$.

It seems that in the case of general Dirichlet series no ’hypercontractivity results’ are available, and so our proof of Theorem 4.10 in [25] follows an entirely different approach. Nevertheless the following problem seems interesting in itself.

Problem 4.15**.**

Given $D=\sum a_{n}e^{\lambda_{n}s}\in\mathcal{H}_{p}(\lambda),1\leq p<\infty$. Under which assumptions on $\lambda$ is it true that $D_{u}=\sum a_{n}e^{\lambda_{n}u}e^{\lambda_{n}s}\in\mathcal{H}_{q}(\lambda)$ for all $1\leq q<\infty$ and $u>0$?

From [26, §12] we know that for $1\leq p<\infty$

[TABLE]

The second equality was implicitly proved in [3], and in view of the coefficient preserving identity $\mathcal{H}_{p}=H_{p}(\mathbb{T}^{\infty})$, it implies that the infimum over all $u>0$ such that for each $f\in H_{p}(\mathbb{T}^{\infty})$ the series

[TABLE]

converges (here we of course talk about absolutele convergence) equals $1/2$ (compare also with (10)). The next result shows that there is no such ’wall’ if we order the sum ’blockwise according to the order of $\mathbb{N}$’.

Corollary 4.16**.**

Let $f\in H_{p}(\mathbb{T}^{\infty}),1\leq p<\infty$. Then for all $u>0$

[TABLE]

and moreover

[TABLE]

where $C=C(u)$ only depends on $u$.

What about the case $\omega=1$ (the neutral element of $G$) in Theorem 4.10, i.e. (pointwise) convergence of Dirichlet series from $\mathcal{H}_{p}(\lambda)$? By $(\ref{Bohrforpresident})$ we always have $\sigma_{c}(D)\leq\sigma_{a}(D)\leq L(\lambda)$ for all $D\in\mathcal{H}_{p}(\lambda)$ (since in this case the sequence $(a_{n}(D))$ of Dirichlet coefficients is always bounded). In the particular case $\lambda=(\log n)$, we know from (26) that

[TABLE]

Problem 4.17**.**

Given a frequency $\lambda$, give reasonable upper and lower estimates for $\sup_{D\in\mathcal{H}_{p}(\lambda)}\sigma_{c}(D)$ and $\sup_{D\in\mathcal{H}_{p}(\lambda)}\sigma_{c}(D)$ , for example in terms of $L(\lambda)$.

4.8. Carleson’s theorem

What happens, if we consider the case $u=0$ in the preceding two sections, in particular in Theorem 4.8?

It turns out that this question is intimately related with one of the most famous theorems in Fourier analysis, namely Carleson’s theorem: The Fourier series of every $f\in L_{2}(\mathbb{T})$ converges almost everywhere. More generally, the Carleson-Hunt theorem states that if $f\in L_{p}(\mathbb{T})$, $1<p<\infty$, then $f(z)=\sum_{n}\widehat{f}(n)z^{n}$ for almost all $z\in\mathbb{T}$ and

[TABLE]

Recall that a frequency is of integer type if it has a basis $B$ for which the Bohr matrix $R$ only contains integers. For $\lambda$’s of this type the following theorem extends the Carleson-Hunt theorem to our setting of $\lambda$-Dirichlet groups, and for such $\lambda$’s it is a strong improvement of Theorem 4.3.

Theorem 4.18**.**

Let $\lambda$ be a frequency of integer type, $(G,\beta)$ a $\lambda$-Dirichlet group, and $1<p<\infty$. Then for every $f\in H_{p}^{\lambda}(G)$

[TABLE]

almost everywhere on $G$, and

[TABLE]

where $C=C(p)$ is a constant which only depends on $p$.

The case $p=2$ is due to [34, Theorem 1.4], whereas the proof of the more general case given here follows closely the ideas of this article combining them with the Carleson-Hunt theorem and a magic trick of Fefferman from [30]. This is carried out in [25].

It is somewhat surprising that Theorem 4.18 doesn’t require $(BC)$ for $\lambda$.

Problem 4.19**.**

Does Theorem 4.8 hold for a wider class of frequencies than the class of $\lambda$’s satisfying $(BC)$? Is it even true without any condition on $\lambda$?

Theorem 4.18 easily translates into a sort of Carleson-Hunt theorem for functions on the polytorus $\mathbb{T}^{\infty}$. Obviously, given some $f\in H_{p}(\mathbb{T}^{\infty})$ with $1<p<\infty$, there in general is no $z\in\mathbb{T}^{\infty}$ such that $f(z)=\sum_{\alpha}\hat{f}(\alpha)z^{\alpha}$.

Corollary 4.20**.**

Let $f\in H_{p}(\mathbb{T}^{\infty})$ and $1<p<\infty$. Then for almost all $z\in\mathbb{T}^{\infty}$

[TABLE]

and moreover,

[TABLE]

5. Vector-valued aspects of general Dirichlet series

We now briefly discuss some results on vector-valued general Dirichlet series $\sum a_{n}e^{-\lambda_{n}s}$, where $(a_{n})\subset X$ for some Banach space $X$. All results are related to the above topics on general Dirichlet series with scalar coefficients, and most of them are taken from [16].

In the ordinary case $\lambda=(\log n)$ the study of functional analytic aspects of vector-valued Dirichlet series accumulated quite some amount of research, see e.g. [14], [15], [17], [18], [20], [21], [22], or [23]. As an example we recall the following strong extension of (2) from [20]: The maximal width of the strip of uniform, non absolute convergence of $X$-valued ordinary Dirichlet series is given by the formula

[TABLE]

where $\text{cot}(X)$ denotes the infimum over all $2\leq q\leq\infty$ for which $X$ has cotype $q$ (see also [26, Theorem 26.4]).

Given a Banach space $X$ and a frequency $\lambda$, we denote by $\mathcal{D}(\lambda,X)$ the space of all $\lambda$-Dirichlet series with coefficients in $X$, and by $\mathcal{D}_{\infty}(\lambda,X)$ the space of all $D\in\mathcal{D}(\lambda,X)$ which on $[Re>0]$ define bounded (and then necessarily holomorphic) functions. Moreover, the by now obvious definition of the a priori larger space $\mathcal{D}_{\infty}^{ext}(\lambda,X)$ is then clear.

The Hahn-Banach theorem transports many results from scalar-valued Dirichlet series to vector-valued ones. For instance this way we may easily deduce from Corollary 2.7 that $\left(\mathcal{D}_{\infty}(\lambda,X),\|\cdot\|_{\infty}\right)$ is a normed space with

[TABLE]

for all $D\in\mathcal{D}_{\infty}(\lambda,X)$. In [16] it is shown that $\mathcal{D}_{\infty}(\lambda)$ is complete if and only if $\mathcal{D}_{\infty}(\lambda,X)$ is.

Another straight forward application of the Hahn-Banach theorem yields that the vector-valued version of Theorem 2.4 holds, and consequently also the qualitative versions (16) and (17) of Bohr’s theorem (where in all inequalities the absolute value of course has to be replaced by the norm in $X$).

Given $1\leq p\leq\infty$, $X$ a Banach space and $\lambda$ a frequency, the definition of Hardy spaces of $X$-valued Dirichlet series extends naturally to the vector-valued situation. As in the scalar case we for any $\lambda$-Dirichlet group $(G,\beta)$ define the Banach space

[TABLE]

and consider the vector-valued Bohr map

[TABLE]

Then it again turns out that the Hardy space

[TABLE]

together with the norm $\|D\|_{p}:=\|f\|_{p}$ of all $X$-valued $\lambda$-Dirichlet series, forms a Banach space which is independent of the chosen $\lambda$-Dirichlet group.

In this final section of our survey we focus on two different topics from [16]. In the first one we ask up to which extend Theorem 4.1 transfers to the vector-valued setting? More precisely, for which Banach spaces $X$ and frequencies $\lambda$ does the equality $\mathcal{D}_{\infty}(\lambda,X)=\mathcal{H}_{\infty}(\lambda,X)$ hold (with an coefficient preserving isometry). Secondly, we revisit the maximal inequalities from the Sections 4.6 and 4.8 in the vector-valued case.

5.1. Coincidence of vector-valued $\boldsymbol{H_{\infty}}$’s

We recall that a Banach space $X$ has the analytic Radon Nikodym property (short ARNP) if every $f\in H_{\infty}(\mathbb{D},X)$ has radial limits in almost all $w\in\mathbb{T}$. This happens if and only if $H_{\infty}(\mathbb{T},X)=H_{\infty}(\mathbb{D},X)$ (via an isometry identifying Fourier and monomial coefficients). Then it is straight forward to see that $\mathcal{D}_{\infty}((n),X)=\mathcal{H}_{\infty}((n),X)$ if and only if $X$ has ARNP.

The main result of [27] (see also [26, Theorem 24.17]) shows that this result holds true for ordinary Dirichlet series, so for the frequency $\lambda=(\log n)$, and in the following theorem we state an extension of all this for general Dirichlet series which is proven in [16].

Theorem 5.1**.**

Let $\lambda$ be any frequency and $X$ a non trivial Banach space with ARNP. Then $\mathcal{D}_{\infty}(\lambda)=\mathcal{H}_{\infty}(\lambda)$ if and only if

[TABLE]

In particular, (27) holds whenever $X$ has ARNP and $\lambda$ satisfies $(BC)$.

But in fact there are Banach spaces $X$ without ARNP as well as frequencies $\lambda$ for which $\mathcal{D}_{\infty}(\lambda)=\mathcal{H}_{\infty}(\lambda)$, such that (27) is no equality. To see this we extend Theorem 2.9 to the vector-valued case. Denote by $\ell_{1}^{w}(X)$ the Banach space of all weakly summable $X$-valued sequences $(a_{n})$ with norm

[TABLE]

and by $\ell^{w,0}_{1}(X)$ the closed subspace of $\ell_{1}^{w}(X)$ consisting of all sequences $(a_{n})$ which are unconditionally summable. Recall that $\ell^{w,0}_{1}(X)=\ell^{w}_{1}(X)$ if and only if $X$ does not contain an isomorphic copy of $c_{0}$. The following theorem is the announced extension of Theorem 2.9 from [16].

Theorem 5.2**.**

If $\lambda$ is $\mathbb{Z}$-linearly independent, then for every Banach space $X$ the following isometric inclusions hold:

[TABLE]

In particular, $\mathcal{D}_{\infty}(\lambda,X)=\mathcal{H}_{\infty}(\lambda,X)$ holds isometrically whenever $c_{0}$ is no isomorphic subspace of $X$.

Banach lattices have ARNP if and only if they don’t contain an isomorphic copy of $c_{0}$. But in contrast to this, there are Banach spaces which fail to have ARNP as well as an isomorphic copy of $c_{0}$ (see [13, §4]). Consequently, Theorem 5.2 shows that if $\lambda$ is $\mathbb{Z}$-linearly independent, then the equality $\mathcal{H}_{\infty}(\lambda,X)=\mathcal{D}_{\infty}(\lambda,X)$ does not necessarily imply that $X$ has ARNP.

5.2. Maximal inequalities - vector-valued

Now we discuss the inequalities from the Sections 4.2, 4.6 and 4.8. Recall that our proof of Theorem 4.3 relies explicitely on the boundedness of the Riesz projection on $L_{p}(G)$, where $G$ is some Dirichlet group. This fact remains valid for $X$-valued functions whenever $X$ is a so-called UMD-space which, as observed in [16], is an immediate consequence of results from [2].

Theorem 5.3**.**

Let $G$ be a Dirichlet group, $X$ a Banach space and $1<p<\infty$. Then $X$ has UMD if and only if the Riesz projection

[TABLE]

is bounded.

As done in [24] for the scalar case, a standard argument leads to the following consequence generalizing Theorem 4.3 (see once again [16]).

Corollary 5.4**.**

For every frequency $\lambda$ and every UMD-space $X$ we have

[TABLE]

Let us finally again turn to the Carleson-Hunt theorem. The proof of Theorem 4.18 given in [25], extends in the following sense from scalar to vector-valued functions.

Theorem 5.5**.**

Assume that $X$ is a Banach space for which the Carleson-Hunt maximal inequality holds, i.e. for every $1<p<\infty$ there is some $C>0$ such that for every $f\in L_{p}(\mathbb{T},X)$

[TABLE]

Then for every frequency $\lambda$ of integer type, every $\lambda$-Dirichlet group $(G,\beta)$, and every $1<p<\infty$ we have that for every $f\in H_{p}^{\lambda}(G,X)$

[TABLE]

almost everywhere on $G$, and

[TABLE]

where $C=C(p)$ is a constant which only depends on $p$.

In particular, if $X$ is a Banach lattice with UMD, then (28) holds true (see e.g. [31]), hence in this case its $X$-valued variant (29) is valid. Actually, it is proven in [16] that for $\mathbb{Z}$-linearly independent frequencies $\lambda$ the assumption UMD on $X$ in fact is superfluous.

Proposition 5.6**.**

If $\lambda$ is $\mathbb{Z}$-linearly independent, then (29) holds for all Banach spaces $X$.

Since the proof of Theorem 4.8 needs for example tools like the Hausdorff-Young inequality, there so far seems no satisfying answer to the following final problem.

Problem 5.7**.**

Under which assumptions on the Banach space $X$ and the frequency $\lambda$ for all $f\in H_{p}^{\lambda}(G,X)$ and $u>0$

[TABLE]

If there is such an estimate, one may ask on which of the parameters $\lambda,X,p$ and $u$ the constant $C$ actually depends. Recall that in Theorem 4.8, which deals with the scalar case $X=\mathbb{C}$ and $\lambda$’s with $(BC)$, the constant $C$ in $(\ref{ineqmaxi})$ does not depend on $p$ (but on $u$ and $\lambda$). This is a useful fact since by Remark 4.11, Theorem 4.8 then implies Bohr’s theorem for $\lambda$’s with $(BC)$. Since Bohr’s theorem in the vector-valued case holds for every Banach space $X$ provided that $\lambda$ has $(LC)$, Problem 5.7 may hold for every Banach space $X$ and $\lambda$’s satisfying $(LC)$ with some constant $C$ independent of $p$. If we ask for a constant $C$ depending on $p$, then in view of Theorem 5.5 it may happen that Problem 5.7 has an affirmative answer for every frequency $\lambda$ but with restrictions on the Banach space $X$.