# Hardy spaces of general Dirichlet series – a survey

Andreas Defant  Andreas Defant
Institut für Mathematik,
Carl von Ossietzky Universität,
26111 Oldenburg, Germany<EMAIL_ADDRESS>and  Ingo
Schoolmann  Ingo Schoolmann
Institut für Mathematik,
Carl von Ossietzky Universität,
26111 Oldenburg, Germany<EMAIL_ADDRESS>
###### 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.

††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

$\sigma_{c}(D)\leq\sigma_{b}(D)\leq\sigma_{u}(D)\leq\sigma_{a}(D)\,.$

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

(1) $\sigma_{b}(D)=\sigma_{u}(D)\,,$

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

$S=\sup\\{\sigma_{a}(D)-\sigma_{u}(D)\colon D=\sum a_{n}n^{-s}\\}\,,$

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

(2) $S=\dfrac{1}{2}$

(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

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

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

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

which preserves Fourier- and Dirichlet coefficients in the sense that

(4) $\hat{f}(\alpha)=a_{n}\,\,\text{ whenever
}n=\mathfrak{p}^{\alpha}:=p_{1}^{\alpha_{1}}\ldots p_{N}^{\alpha_{N}}\,,$

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

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

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

(5) $\mathbb{T}^{\infty}$$\mathbb{T}$$\mathbb{R}$$z^{\alpha}$$e^{-it\log
n}$$\beta$

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

(6)
$H_{\infty}(\mathbb{T}^{\infty})=H_{\infty}(B_{c_{0}})\,,\leavevmode\nobreak\
f\mapsto g$

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

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

(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):

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

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

(8)
$\mathcal{D}_{\infty}=\mathcal{H}_{\infty}=H_{\infty}(\mathbb{T}^{\infty})\,.$

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

$H_{p}(\ell_{2}\cap B_{c_{0}})$

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

(9) $\|g\|_{H_{p}(\ell_{2}\cap
B_{c_{0}})}=\sup_{n\in\mathbb{N}}\sup_{0<r<1}\left(\int_{\mathbb{T}^{n}}|g(rw_{1},\ldots,rw_{n},0,0,\ldots)|^{p}d(w_{1},\ldots,w_{n})\right)^{\frac{1}{p}}<\infty\,.$

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

(10) $H_{p}(\ell_{2}\cap B_{c_{0}})=H_{p}(\mathbb{T}^{\infty})\,.$

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

$\sum a_{n}e^{-\lambda_{n}s}$

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’

$\displaystyle\sigma_{c}(D)=\inf\left\\{\sigma\in\mathbb{R}\mid D\text{
converges on }[Re>\sigma]\right\\},$
$\displaystyle\sigma_{a}(D)=\inf\left\\{\sigma\in\mathbb{R}\mid D\text{
converges absolutely on }[Re>\sigma]\right\\},$
$\displaystyle\sigma_{u}(D)=\inf\left\\{\sigma\in\mathbb{R}\mid D\text{
converges uniformly on }[Re>\sigma]\right\\},$

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 $D=\sum a_{n}e^{-\lambda_{n}s}$,

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

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:

(11)
$\sigma_{u}(D)\leq\limsup_{N}\frac{\log\Big{(}\sup_{t\in\mathbb{R}}\big{|}\sum^{N}_{n=1}a_{n}e^{-\lambda_{n}it}\big{|}\Big{)}}{\lambda_{N}},$

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

$\sigma_{b}(D)=\inf\sigma\,,$

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
$\sigma_{c}(D)<\infty$,

$\displaystyle\sigma_{b}^{ext}(D)=\inf\sigma\,,$

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

(12) $\sigma_{b}^{ext}(D)=\sigma_{u}(D)$

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

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

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 $(BC)$):

(13) $\exists\leavevmode\nobreak\ l=l(\lambda)>0\leavevmode\nobreak\
\forall\leavevmode\nobreak\ \delta>0\leavevmode\nobreak\
\exists\leavevmode\nobreak\ C>0\leavevmode\nobreak\ \forall
n\in\mathbb{N}:\leavevmode\nobreak\ \leavevmode\nobreak\
\lambda_{n+1}-\lambda_{n}\geq Ce^{-(l+\delta)\lambda_{n}};$

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:

(14) $\forall\leavevmode\nobreak\ \delta>0\leavevmode\nobreak\
\exists\leavevmode\nobreak\ C>0\leavevmode\nobreak\
\forall\leavevmode\nobreak\ n\in\mathbb{N}\colon\leavevmode\nobreak\
\leavevmode\nobreak\ \lambda_{n+1}-\lambda_{n}\geq
Ce^{-e^{\delta\lambda_{n}}}.$

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

$\exists\leavevmode\nobreak\ x>0\leavevmode\nobreak\
\exists\leavevmode\nobreak\ C>0\leavevmode\nobreak\
\forall\leavevmode\nobreak\ n\in\mathbb{N}\colon\leavevmode\nobreak\
\leavevmode\nobreak\ \lambda_{n+1}-\lambda_{n}\geq Ce^{-e^{x\lambda_{n}}}.$

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 $N\geq 2$

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

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

$S_{N}\colon\mathcal{D}_{\infty}^{ext}(\lambda)\to\mathcal{D}_{\infty}(\lambda),\leavevmode\nobreak\
\leavevmode\nobreak\ D\mapsto\sum_{n=1}^{N}a_{n}(D)e^{-\lambda_{n}s}\,.$

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 $N\in\mathbb{N}$:

$\sup_{[Re>0]}\left|\sum_{n=1}^{N}a_{n}e^{-\lambda_{n}s}\right|\leq
C\frac{\Gamma(k+1)}{k}\left(\frac{\lambda_{N+1}}{\lambda_{N+1}-\lambda_{N}}\right)^{k}\|D\|_{\infty},$

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

(16)
$\|S_{N}\colon\mathcal{D}_{\infty}^{ext}(\lambda)\to\mathcal{D}_{\infty}(\lambda)\|\leq
C_{1}(\lambda)\lambda_{N},$

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

(17)
$\|S_{N}\colon\mathcal{D}_{\infty}^{ext}(\lambda)\to\mathcal{D}_{\infty}(\lambda)\|\leq
C_{2}(\lambda,\delta)e^{\delta\lambda_{N}};$

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

$S_{N}\colon H_{\infty}(\mathbb{T})\to
H_{\infty}(\mathbb{T})\,,\,\,\,S_{N}(f)=\sum_{n=0}^{N}\hat{f}(n)z^{n}$

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

$R_{x}^{k}(D)=\sum_{\lambda_{n}<x}a_{n}\left(1-\frac{\lambda_{n}}{x}\right)^{k}e^{-\lambda_{n}s}$

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

(18) $\sup_{x\geq
0}\|R_{x}^{k}(D)\|_{\infty}\leq\frac{e}{2\pi}\frac{\Gamma(k+1)}{k}\|D\|_{\infty}.$

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

$a_{n}(D)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(\sigma+it)e^{(\sigma+it)\lambda_{n}}$

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

$L(\lambda):=\limsup_{N\to\infty}\frac{\log(N)}{\lambda_{N}}=0.$

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

(19) $L(\lambda)=\sigma_{c}\left(\sum
e^{-\lambda_{n}s}\right)=\sigma_{a}\left(\sum
e^{-\lambda_{n}s}\right)=\sup_{D\in\mathcal{D}(\lambda)}\sigma_{a}(D)-\sigma_{c}(D).$

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

$\sigma_{b}^{ext}(D)=\sigma_{a}(D)$

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,

$\mathcal{D}^{ext}_{\infty}(\lambda)=\mathcal{D}_{\infty}(\lambda)=\ell_{1}\,,\,\,\,\,\,\sum
a_{n}e^{-\lambda_{n}s}\mapsto(a_{n})\,,$

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

$\sup_{N\in\mathbb{N}}\sup_{[Re>0]}\left|\sum_{n=1}^{N}a_{n}e^{-\lambda_{n}s}\right|=\|D\|_{\infty}.$

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

(20) $\left\\{\lambda_{n}+\frac{j}{s_{n}}(\lambda_{n+1}-\lambda_{n})\mid
n\in\mathbb{N},\leavevmode\nobreak\ j=0,\ldots s_{n}-1\right\\}$

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

(21) $\widehat{G}=\\{h_{x}\mid x\in\widehat{\beta}(\widehat{G})\\}.$

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 $n\in\mathbb{N}$:

$G$$\mathbb{T}$$\mathbb{R}$$h_{\lambda_{n}}$$e^{-i\lambda_{n}\cdot}$$\beta$

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

$\beta_{\mathbb{T}^{\infty}}\colon\mathbb{R}\to\mathbb{T}^{\infty},\leavevmode\nobreak\
\leavevmode\nobreak\ t\mapsto\mathfrak{p}^{-it}$

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

$\beta_{\widehat{(U,d)}}\colon\mathbb{R}\to\widehat{(U,d)},\leavevmode\nobreak\
\leavevmode\nobreak\ t\mapsto\left[u\mapsto e^{-itu}\right]$

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

$\beta_{\mathbb{T}}:\mathbb{R}\rightarrow\mathbb{T},\,t\mapsto e^{-it}.$

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

$\beta_{\overline{\mathbb{R}}}\colon(\mathbb{R},|\cdot|)\hookrightarrow\overline{\mathbb{R}},\,\,x\mapsto\left[t\mapsto
e^{-ixt}\right]\,.$

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

(22)
$T_{B}:\bigoplus_{n=1}^{N}\mathbb{Q}\hookrightarrow\mathbb{R},\leavevmode\nobreak\
\leavevmode\nobreak\ \alpha\mapsto\sum\alpha_{j}b_{j}$

is an injective homomorphism, and hence its dual map

$\widehat{T_{B}}\colon\mathbb{R}\to\widehat{\bigoplus_{n=1}^{\infty}\mathbb{Q}_{d}},\leavevmode\nobreak\
\leavevmode\nobreak\ t\mapsto\left[(q_{j})_{j}\mapsto e^{-it\sum
q_{j}b_{j}}\right]$

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

$H_{p}^{\lambda}\left(G\right):=\left\\{f\in L_{p}(G)\mid\text{
$\hat{f}:\widehat{G}\to\mathbb{C}$ is supported by all
$h_{\lambda_{n}},n\in\mathbb{N}$}\right\\}\,,$

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

$\mathcal{B}\colon
H^{\lambda}_{p}(G)\hookrightarrow\mathcal{D}(\lambda),\,\,\,f\sim\sum_{\gamma\in\widehat{G}}\hat{f}(\gamma)\gamma\,\,\,\mapsto\,\,\,\sum_{n\in\mathbb{N}}\widehat{f}(h_{\lambda_{n}})e^{-\lambda_{n}s}$

gives an isometric onto isomorphism

$\mathcal{H}_{p}(\lambda)=H_{p}^{\lambda}(G)\,.$

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

$H_{p}^{\log(n)}(\overline{\mathbb{R}})=\mathcal{H}_{p}(\log(n))=H_{p}(\mathbb{T}^{\infty}),\leavevmode\nobreak\
\leavevmode\nobreak\ \|f\|_{p}=\|D\|_{p}=\|g\|_{p},$

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

$\|D\|_{p}:=\left(\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}\left|\sum^{N}_{n=1}a_{n}e^{-\lambda_{n}it}\right|^{p}dt\right)^{\frac{1}{p}}.$

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

$\psi\colon\mathcal{H}_{p}(\lambda)\to
H_{p}^{R}(\mathbb{T}^{N}),\leavevmode\nobreak\ \leavevmode\nobreak\ D\mapsto
g$

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

$\psi\colon\mathcal{H}_{p}(\lambda)\hookrightarrow\mathcal{H}_{p},\leavevmode\nobreak\
\leavevmode\nobreak\ \sum a_{n}e^{-\lambda_{n}s}\to\sum b_{n}n^{-s}$

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

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

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

$\mathcal{D}_{\infty}(\lambda)=\mathcal{H}_{\infty}(\lambda)\,.$

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

$\mathcal{D}^{ext}_{\infty}(\lambda)\subset\mathcal{H}_{2}(\lambda)\,,$

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

$\widehat{G}=\\{h_{x}\mid x\in\widehat{\beta}(\widehat{G})\\}\,.$

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’

$\Phi\left(\sum a_{k}h_{x_{k}}\right):=\sum_{x_{k}\geq 0}a_{k}h_{x_{k}}$

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

$S_{N}^{p}:\mathcal{H}_{p}(\lambda)\to\mathcal{H}_{p}(\lambda),\,\,\sum
a_{n}e^{-\lambda_{n}s}\mapsto\sum_{n=1}^{N}a_{n}e^{-\lambda_{n}s}$

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 $N\geq 2$

$\|S_{N}^{1}:\mathcal{H}_{1}(\lambda)\to\mathcal{H}_{1}(\lambda)\|\leq
C\lambda_{N}\,,$

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

$\|S_{N}^{1}:\mathcal{H}_{1}(\lambda)\to\mathcal{H}_{1}(\lambda)\|\leq
De^{\delta\lambda_{N}}\,.$

###  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

$H_{1}^{\lambda}(G)\to M_{\lambda}(G),\leavevmode\nobreak\
\leavevmode\nobreak\ f\mapsto f\leavevmode\nobreak\ dm$

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 $\varepsilon$

$D^{N_{k}}_{\varepsilon}=\sum
a_{n}(D^{N_{k}})e^{-\varepsilon\lambda_{n}}e^{-\lambda_{n}s}$

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

$\Phi\colon\mathcal{H}_{p}(\lambda)\hookrightarrow\mathcal{D}_{\infty}(\lambda,\mathcal{H}_{p}(\lambda)),\leavevmode\nobreak\
\leavevmode\nobreak\ \sum
a_{n}e^{-\lambda_{n}s}\mapsto\sum\left(a_{n}e^{-\lambda_{n}s}\right)e^{-\lambda_{n}z}$

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

$\sum_{n}\hat{f}(h_{\lambda_{n}})e^{-\lambda_{n}u}h_{\lambda_{n}}$

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

$P_{u}(t):=\frac{1}{\pi}\frac{u}{u^{2}+t^{2}}\colon\mathbb{R}\to\mathbb{R}$

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

$T_{u}\colon C(G)\to\mathbb{C},\leavevmode\nobreak\ \leavevmode\nobreak\
g\mapsto\int_{\mathbb{R}}(g\circ\beta)(t)P_{u}(t)dt$

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 $u>0$

$f\ast
p_{u}=\sum_{n=1}^{\infty}\widehat{f}(h_{\lambda_{n}})e^{-\lambda_{n}u}h_{\lambda_{n}}$

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

(23)
$\bigg{\|}\sup_{N}\Big{|}\sum_{n=1}^{N}\widehat{f}(h_{\lambda_{n}})e^{-\lambda_{n}u}h_{\lambda_{n}}\Big{|}\bigg{\|}_{p}\leq
C\|f\|_{p}.$

###  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

$D^{\omega}(s)=\sum a_{n}h_{\lambda_{n}}(\omega)e^{-\lambda_{n}s}$

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’

$D_{i\tau_{k}}=\sum a_{n}e^{-i\lambda_{n}\tau_{k}}e^{-\lambda_{n}s}$

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

$D^{\omega}(s)=\sum
a_{n}(D)h_{\lambda_{n}}(\omega)e^{-\lambda_{n}s},\,\omega\in G,$

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

(24)
$\bigg{(}\int_{G}\sup_{N}\Big{|}\sum_{n=1}^{N}a_{n}h_{\lambda_{n}}(\omega)e^{-u\lambda_{n}}\Big{|}^{p}dm(\omega)\bigg{)}^{\frac{1}{p}}\leq
C\|D\|_{p}.$

######  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

$f_{\omega}*P_{u}(t):=\int_{\mathbb{R}}f_{\omega}(t-y)P_{u}(y)dy$

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 $t\in\mathbb{R}$

$D^{\omega}(u+it)=(f_{\omega}*P_{u})(t).$

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

(25) $\displaystyle\iota\colon\Xi\to\mathbb{T}^{\infty},\leavevmode\nobreak\
\leavevmode\nobreak\ \chi\mapsto(\chi(p_{n}))_{n},$

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 $\sum a_{n}n^{-s}\in\mathcal{H}_{p}(\lambda)$

$\left(\int_{\Xi}\sup_{N}\left|\sum_{n=1}^{N}a_{n}\chi(n)n^{-u}\right|^{p}d\chi\right)^{\frac{1}{p}}\leq
C\|D\|_{p}\,.$

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 $\mathbb{T}$

$p_{r}(w):=\sum_{n=-\infty}^{\infty}r^{|n|}w^{n},\leavevmode\nobreak\
\leavevmode\nobreak\ 0<r<1,$

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

$\mathcal{P}_{r}\colon H_{p}(\mathbb{T})\to
H_{q}(\mathbb{T}),\leavevmode\nobreak\ \leavevmode\nobreak\ f\mapsto
f*p_{r}(\omega)=\sum_{n=0}^{\infty}\widehat{f}(n)(rw)^{n}.$

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

$\mathcal{P}_{u}\colon\mathcal{H}_{p}\mapsto\mathcal{H}_{q},\leavevmode\nobreak\
\leavevmode\nobreak\ D\mapsto D_{u}=\sum a_{n}(D)n^{-u}n^{-s}$

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 $1\leq p<\infty$

(26)
$\sup_{D\in\mathcal{H}_{p}}\sigma_{c}(D)=\sup_{D\in\mathcal{H}_{p}}\sigma_{a}(D)=\frac{1}{2}$

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

$\sum_{\alpha}\hat{f}(\alpha)\,\Big{(}\frac{z}{\mathfrak{p}^{u}}\Big{)}^{\alpha}$

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 $u>0$

$\lim_{N}\sum_{\mathfrak{p}^{\alpha}\leq
N}\hat{f}(\alpha)\,\Big{(}\frac{z}{\mathfrak{p}^{u}}\Big{)}^{\alpha}\,\,\,\,\,\text{a.e.
on $\mathbb{T}^{\infty}$}\,,$

and moreover

$\bigg{(}\int_{\mathbb{T}^{\infty}}\sup_{N}\Big{|}\sum_{\mathfrak{p}^{\alpha}\leq
N}c_{\alpha}\Big{(}\frac{w}{\mathfrak{p}^{u}}\Big{)}^{\alpha}\Big{|}^{p}d\omega\bigg{)}^{1/p}\leq
C(u)\|f\|_{p}\,,$

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

$\sup_{D\in\mathcal{H}_{p}}\sigma_{c}(D)=\sup_{D\in\mathcal{H}_{p}}\sigma_{a}(D)=\frac{L((\log
n))}{2}.$

######  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

$\Big{\|}\sup_{N}\big{|}\sum_{n=-N}^{N}\hat{f}(n)z^{n}\big{|}\Big{\|}_{p}\leq
C\|f\|_{p}\,,$

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 $f\in
H_{p}^{\lambda}(G)$

$f=\sum_{n=1}^{\infty}\hat{f}(h_{\lambda_{n}})h_{\lambda_{n}}$

almost everywhere on $G$, and

$\bigg{\|}\sup_{N}\Big{|}\sum_{n=1}^{N}\hat{f}(h_{\lambda_{n}})h_{\lambda_{n}}\Big{|}\bigg{\|}_{p}\leq
C\|f\|_{p}\,,$

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

$f(z)=\lim_{N}\sum_{\mathfrak{p}^{\alpha}\leq N}\hat{f}(\alpha)z^{\alpha}\,,$

and moreover,

$\bigg{\|}\sup_{N}\Big{|}\sum_{\mathfrak{p}^{\alpha}\leq
N}\hat{f}(\alpha)z^{\alpha}\Big{|}^{p}\bigg{\|}_{p}\leq C(p)\|f\|_{p}\,.$

##  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

$S(X)=1-\frac{1}{\text{cot}(X)}\,,$

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

$\sup_{n\in\mathbb{N}}\|a_{n}\|_{X}\leq\|D\|_{\infty}$

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

$H^{\lambda}_{p}(G,X):=\left\\{f\in L_{p}(G,X)\mid\text{$f:\hat{G}\to X$ is
supported on all $h_{\log n},n\in\mathbb{N}$}\right\\},$

and consider the vector-valued Bohr map

$\mathcal{B}\colon
H_{p}^{\lambda}(G,X)\hookrightarrow\mathcal{D}(\lambda,X),\leavevmode\nobreak\
\leavevmode\nobreak\ f\mapsto
D=\sum\hat{f}(h_{\lambda_{n}})e^{-\lambda_{n}s}\,.$

Then it again turns out that the Hardy space

$\mathcal{H}_{p}(\lambda,X):=\mathcal{B}(H_{p}^{\lambda}(G,X))$

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

(27) $\mathcal{D}_{\infty}(\lambda,X)=\mathcal{H}_{\infty}(\lambda,X)\,.$

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

$w((a_{n}))=\sup_{x^{\prime}\in
B_{X^{\prime}}}\sum_{n=1}^{\infty}|x^{\prime}(a_{n})|\,,$

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:

$\ell^{w,0}_{1}(X)\subset\mathcal{H}_{\infty}(\lambda,X)\subset\mathcal{D}_{\infty}(\lambda,X)\subset\ell^{w}_{1}(X)$

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

$R\colon L_{p}(G,X)\to L_{p}(G,X),\leavevmode\nobreak\ \leavevmode\nobreak\
f\mapsto\sum_{x\geq 0}\widehat{f}(h_{x})h_{x}$

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

$\sup_{N}\Big{\|}S_{N}^{p,X}:\mathcal{H}_{p}(\lambda,X)\to\mathcal{H}_{p}(\lambda,X),\,\,\sum
a_{n}e^{-\lambda_{n}s}\mapsto\sum_{n=1}^{N}a_{n}e^{-\lambda_{n}s}\Big{\|}<\infty.$

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 $f\in L_{p}(\mathbb{T},X)$

(28)
$\bigg{\|}\sup_{N}\Big{\|}\sum_{n=-N}^{N}\hat{f}(n)z^{n}\Big{\|}\bigg{\|}_{L_{p}(\mathbb{T})}\leq
C\|f\|_{L_{p}(\mathbb{T},X)}\,.$

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

$f=\sum_{n=1}^{\infty}\hat{f}(h_{\lambda_{n}})h_{\lambda_{n}}$

almost everywhere on $G$, and

(29)
$\bigg{\|}\sup_{N}\Big{\|}\sum_{n=1}^{N}\hat{f}(h_{\lambda_{n}})h_{\lambda_{n}}\Big{\|}\bigg{\|}_{p}\leq
C\|f\|_{p}\,,$

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 $u>0$

$\left(\int_{G}\sup_{N}\left\|\sum_{n=1}^{N}\widehat{f}(h_{\lambda_{n}})e^{-u\lambda_{n}}h_{\lambda_{n}}(\omega)\right\|_{X}^{p}d\omega\right)^{\frac{1}{p}}\leq
C\|f\|_{p}\leavevmode\nobreak\ ?$

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 $X$.

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