Riesz means in Hardy spaces on Dirichlet groups

TL;DR

This paper investigates the almost everywhere Riesz summability of Dirichlet series and Fourier series of $H_1$-functions on $ ext{Dirichlet}$ groups, introducing a new maximal operator and applying results to various function spaces.

Contribution

It develops a weak-type Hardy-Littlewood maximal operator for $ ext{Dirichlet}$ groups and studies the convergence of Dirichlet series and Fourier series in this context.

Findings

Almost all vertical limits of $ ext{Dirichlet}$ series are Riesz-summable almost everywhere.

Established a weak-type $(1, ext{infinity})$ maximal operator for $ ext{Dirichlet}$ groups.

Applications to $H_1$-functions on the infinite torus, Dirichlet series, and bounded holomorphic functions.

Abstract

Given a frequency $\lambda=(\lambda_n)$, we study when almost all vertical limits of a $\mathcal{H}_1$-Dirichlet series $\sum a_n e^{-\lambda_ns}$ are Riesz-summable almost everywhere on the imaginary axis. Equivalently, this means to investigate almost everywhere convergence of Fourier series of $H_1$-functions on so-called $\lambda$-Dirichlet groups, and as our main technical tool we need to invent a weak-type $(1, \infty)$ Hardy-Littlewood maximal operator for such groups. Applications are given to $H_1$-functions on the infinite dimensional torus $\mathbb{T}^\infty$, ordinary Dirichlet series $\sum a_n n^{-s}$, as well as bounded and holomorphic functions on the open right half plane, which are uniformly almost periodic on every vertical line.

Equations

$$\hat{G}=\left\{h_x\mid x\in \mathrm{Im}\hat{\beta }\right\}.$$

$$H_p^{\lambda }(G):=\left\{f\in L_p(G)\mid \forall \text{leavevmode}\gamma \in \hat{G}:\text{leavevmode}\hat{f}(\gamma )\ne 0\Rightarrow \gamma =h_{{\lambda }_n}\text{leavevmode}\text{for some n}\right\},$$

$$T:H_p^{\lambda }(G_1)\to H_p^{\lambda }(G_2),\text{leavevmode}\text{leavevmode}f\mapsto g,$$

$$\widehat{f}\Big{(}(\widehat{\beta_{1}})^{-1}(x)\Big{)}=\widehat{g}\Big{(}(\widehat{\beta_{2}})^{-1}(x)\Big{)}.$$

$${\beta }_{\overline{\mathbb{R}}}:\mathbb{R}\hookrightarrow \overline{\mathbb{R}},x\mapsto \left[t\mapsto e^{-ixt}\right],$$

$${\mathbb{T}}^{\infty }:=\prod _{n=1}^{\infty }\mathbb{T}$$

$${\beta }_{{\mathbb{T}}^{\infty }}:\mathbb{R}\to {\mathbb{T}}^{\infty },\text{leavevmode}\text{leavevmode}t\mapsto {\mathfrak{p}}^{-it}=(2^{-it},3^{-it},5^{-it},\dots ),$$

$$H_p({\mathbb{T}}^{\infty })=H_p^{(\log n)}({\mathbb{T}}^{\infty })$$

$$\iota :\Xi \to {\mathbb{T}}^{\infty },\text{leavevmode}\text{leavevmode}\chi \mapsto \chi (\mathfrak{p})=(\chi (p_n){)}_n,$$

$${\beta }_{\Xi }:\mathbb{R}\to \Xi ,\text{leavevmode}\text{leavevmode}t\mapsto [p_k\to \chi (p_k)],$$

$$\Big{(}\int_{G}\sup_{x}\Big{|}\sum_{\lambda_{n}<x}\widehat{f}(h_{\lambda_{n}})h_{\lambda_{n}}(\omega)\Big{|}^{p}d\omega\Big{)}^{\frac{1}{p}}\leq\text{CH}_{p}\,\|f\|_{p}.$$

$$f(z)=\lim_{N\to\infty}\frac{1}{N}\sum_{k=0}^{N-1}\sum_{n\leq k}\widehat{f}(n)z^{n}=\lim_{N\to\infty}\sum_{n=0}^{N-1}\widehat{f}(n)\Big{(}1-\frac{n}{N}\Big{)}z^{n}$$

$$\sum _{\alpha :1\le {\mathfrak{p}}^{\alpha }\le k}\hat{f}(\alpha )z^{\alpha },k\in \mathbb{N},$$

$$f(z)=\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{k=0}^{N-1}\sum _{{\mathfrak{p}}^{\alpha }\le k}\hat{f}(\alpha )z^{\alpha }?$$

$$\lim_{x\to\infty}\sum_{\lambda_{n}<x}\Big{(}1-\frac{\lambda_{n}}{x}\Big{)}^{k}c_{n}$$

$$R_{x}^{\lambda,k}\Big{(}\sum c_{n}\Big{)}:=\sum_{\lambda_{n}<x}\Big{(}1-\frac{\lambda_{n}}{x}\Big{)}^{k}c_{n}$$

$$\sum_{n=1}^{\infty}c_{n}=\lim_{x\to\infty}\sum_{\lambda_{n}<x}\Big{(}1-\frac{\lambda_{n}}{x}\Big{)}^{k}c_{n}.$$

$$\lim_{N\to\infty}\Big{(}\frac{\lambda_{N+1}-\lambda_{N}}{\lambda_{N+1}}\Big{)}^{k}\Big{(}\sum_{n=1}^{N}c_{n}-C\Big{)}=0\,.$$

$$\lim_{x\to\infty}R_{x}^{e^{\lambda},k}\big{(}\sum c_{n}\big{)}=\lim_{x\to\infty}R_{e^{x}}^{e^{\lambda},k}\big{(}\sum c_{n}\big{)}=\lim_{x\to\infty}\sum_{\lambda_{n}<x}\Big{(}1-\frac{e^{\lambda_{n}}}{e^{x}}\Big{)}^{k}c_{n}\,;$$

$$S_{x}^{\lambda,k}\Big{(}\sum c_{n}\Big{)}:=\sum_{\lambda_{n}<x}\Big{(}1-\frac{e^{\lambda_{n}}}{e^{x}}\Big{)}^{k}c_{n}$$

$$\lim_{x\to\infty}\sum_{\lambda_{n}<x}\hat{f}(h_{\lambda_{n}})\Big{(}1-\frac{\lambda_{n}}{x}\Big{)}^{k}h_{\lambda_{n}}(\omega)$$

$$R_{x}^{\lambda,k}(f):=\sum_{\lambda_{n}<x}\hat{f}(h_{\lambda_{n}})\Big{(}1-\frac{\lambda_{n}}{x}\Big{)}^{k}h_{\lambda_{n}}$$

$$S_{x}^{\lambda,k}(f):=\sum_{\lambda_{n}<x}\hat{f}(h_{\lambda_{n}})\Big{(}1-\frac{e^{\lambda_{n}}}{e^{x}}\Big{)}^{k}\,,$$

$$\underset{x\to \infty }{\lim }{R}_x^{e^{\lambda },k}(f)(\omega )=\underset{x\to \infty }{\lim }{R}_{e^x}^{e^{\lambda },k}(f)(\omega )=\underset{x\to \infty }{\lim }{S}_x^{\lambda ,k}(f)(\omega ).$$

$$R_{x}^{\lambda,1}(f)=\sum_{n<x}\widehat{f}(n)\Big{(}1-\frac{n}{x}\Big{)}z^{n},$$

$$R_{x}^{\lambda,k}(f)=\sum_{\log\mathfrak{p}^{\alpha}<x}\widehat{f}(\alpha)\Big{(}1-\frac{\log\mathfrak{p}^{\alpha}}{x}\Big{)}^{k}z^{\alpha},\leavevmode\nobreak\ \leavevmode\nobreak\ x>0,$$

$$\displaystyle R_{x}^{e^{\lambda},k}(f)(z)=\sum_{\mathfrak{p}^{\alpha}<x}\widehat{f}(\alpha)\Big{(}1-\frac{\mathfrak{p}^{\alpha}}{x}\Big{)}^{k}z^{\alpha}\,,$$

$$\displaystyle R_{N}^{e^{\lambda},1}(f)(z)=\sum_{\mathfrak{p^{\alpha}}<N}\hat{f}(\alpha)\big{(}1-\frac{\mathfrak{p}^{\alpha}}{N}\big{)}z^{\alpha}=\frac{1}{N}\sum_{k=0}^{N-1}\sum_{\mathfrak{p^{\alpha}}\leq k}\hat{f}(\alpha)z^{\alpha}.$$

$$f=\underset{x\to \infty }{\lim }{R}_x^{\lambda ,k}(f)$$

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

Full text

††footnotetext: 2018 Mathematics Subject Classification: Primary 43A17, Secondary 30H10, 30B50††footnotetext: Key words and phrases: Fourier series on groups, general Dirichlet series, vertical limits, Riesz summability, Hardy spaces ††footnotetext:

1. Introduction

Let $\lambda$ be a frequency, i.e. a strictly increasing, unbounded sequence of non-negative real numbers. Moreover, let $G$ be a compact abelian group, and $\beta:\mathbb{R}\to G$ a continuous homomorphism of groups with dense range such that for each character $e^{-i\lambda_{n}\cdot}:\mathbb{R}\to\mathbb{T}$ there is a (then unique) character $h_{\lambda_{n}}:G\to\mathbb{T}$ with $e^{-i\lambda_{n}\boldsymbol{\cdot}}=h_{\lambda_{n}}\circ\beta$.

For $1\leq p\leq\infty$ denote by $H^{\lambda}_{p}(G)$ the Hardy space of all $f\in L_{p}(G)$ which have a Fourier transform $\hat{f}:\hat{G}\to\mathbb{T}$ supported by all characters $h_{\lambda_{n}}$. It is known that for $1<p<\infty$ every $f\in H_{p}^{\lambda}(G)$ has an almost everywhere convergent Fourier series representation $f(\omega)=\sum_{n=1}^{\infty}\widehat{f}(h_{\lambda_{n}})h_{\lambda_{n}}(\omega)$.

Inspired by the work [10] of Hardy and Riesz on general Dirichlet series from 1915, we in this article study almost everywhere Riesz-summability of the Fourier series of functions $f\in H^{\lambda}_{1}(G)$. The main tool is given by an appropriate weak-type $(1,\infty)$ Hardy-Littlewood maximal operator.

As a particular case we look at the frequency $\lambda=(\log n)$, the infinite dimensional torus $G=\mathbb{T}^{\infty}$, and the Kronecker flow $\beta:\mathbb{R}\to G,t\mapsto(p_{k}^{it})$, where $p_{k}$ denotes the $k$th prime. Our results prove that each $f\in H_{1}(\mathbb{T}^{\infty})$ almost everywhere is the pointwise limit of its logarithmic Riesz means, whereas for arithmetic Riesz means (Cesàro means) this in general fails.

Most of our results have equivalent formulations in terms of general Dirichlet series $\sum a_{n}e^{-\lambda_{n}s}$. More precisely, vertical limits of Dirichlet series $\sum a_{n}e^{-\lambda_{n}s}$ which belong to the Hardy space $\mathcal{H}_{1}(\lambda)$, are summable by their first Riesz means of any order $k>0$ on the imaginary axis $[\text{Re}=0]$ (and consequently on the right half-plane).

Another application shows, that the Hardy space $H_{\infty}^{\lambda}(G)$ may be identified with the Banach space of all bounded and holomorphic function on $[Re>0]$ which for every $\varepsilon>0$ are uniformly almost periodic on $[Re>\varepsilon]$, preserving the Fourier and Bohr coefficients.

In the following subsections of this introduction we substantiate all this and provide our reader with the needed preliminaries. In Section 2 we summarize all our results, and in Section 3 we prove them.

1.1. Hardy spaces on Dirichlet groups

Let us briefly recall the general framework of Hardy spaces $H_{p}^{\lambda}(G)$ on so-called $\lambda$-Dirichlet groups $(G,\beta)$ from [4].

A pair $(G,\beta)$ of a compact abelian group $G$ and a homomorphism $\beta\colon(\mathbb{R},+)\to G$ is said to be a Dirichlet group, whenever $\beta$ is continuous and has dense range. In this case, the dual map of $\beta$, that is the mapping $\widehat{\beta}\colon\widehat{G}\hookrightarrow\widehat{\mathbb{R}},\leavevmode\nobreak\ \leavevmode\nobreak\ \gamma\to\gamma\circ\beta$, is injective, where $\widehat{G}$ denotes the dual group of $G$. So, using the identification $\widehat{(\mathbb{R},+)}=\mathbb{R}$ (which we do from now on), the dual group of $G$ via $\widehat{\beta}$ can be considered as a subset of $\mathbb{R}$. Moreover, if $x\in\mathbb{R}$ lies in the image of $\widehat{\beta}$, we write $h_{x}:=(\widehat{\beta})^{-1}(x)$ and obtain

[TABLE]

In other words, for $x\in\widehat{\beta}(\widehat{G})$ the characters $t\mapsto e^{-ixt}$ on $(\mathbb{R},+)$ are precisely those, which allow a unique ’extension’ $h_{x}\in\widehat{G}$ such that $h_{x}\circ\beta=e^{-ix\cdot}$. Note that we do not force $\beta$ to be injective.

Given a frequency $\lambda=(\lambda_{n})$, i.e a strictly increasing non-negative real sequence tending to $\infty$, we call the Dirichlet group $(G,\beta)$ a $\lambda$-Dirichlet group whenever $\lambda\subset\widehat{\beta}(\widehat{G})$. Given such a $\lambda$-Dirichlet group $(G,\beta)$ and $1\leq p\leq\infty$, we define Hardy space

[TABLE]

which being a closed subspace of $L_{p}(G)$, is a Banach space. Of course, $L_{p}(G)$ is here formed with respect to the normalized Haar measure $m$ on $G$. Given two $\lambda$-Dirichlet groups $(G_{1},\beta_{1})$ and $(G_{2},\beta_{2})$, a crucial fact is that the spaces $H_{p}^{\lambda}(G_{1})$ and $H_{p}^{\lambda}(G_{2})$ are isometrically isomorphic (see [4, Corollary 3.21]). More precisely, there is an onto isometry

[TABLE]

which preserves the Fourier coefficients, that is for all $x$ we have

[TABLE]

Let us collect a few crucial examples. The Bohr compactification $\overline{\mathbb{R}}:=\widehat{(\mathbb{R},d)}$ of $\mathbb{R}$ ($d$ the discrete topology) together with the embedding

[TABLE]

forms a Dirichlet group, which obviously for any arbitrary frequency $\lambda$ serves as a $\lambda$-Dirichlet group.

There are two basic examples which later in many more general situations will help us to keep orientation. Consider the frequency $\lambda=(n)=(0,1,2,\ldots)$. Then $G:=\mathbb{T}$ together with $\beta_{\mathbb{T}}(t):=e^{-it}$ is a $(n)$-Dirichlet group, and $\mathcal{H}_{p}((n))$ equals the classical Hardy space $H_{p}(\mathbb{T})$. The second crucial example is $\lambda=(\log n)$. In this case, denoting by $\mathfrak{p}=(p_{n})$ the sequence of prime numbers, the infinite dimensional torus

[TABLE]

(with its natural group structure) together with the so-called Kronecker flow

[TABLE]

gives a $(\log n)$-Dirichlet group. Then $f\in H_{p}^{(\log n)}(\mathbb{T}^{\infty})$ if and only if $f\in L_{p}(\mathbb{T}^{\infty})$ and $\hat{f}(\alpha)=0$ for any finite sequence $\alpha=(\alpha_{k})$ of integers with $\alpha_{k}<0$ for some $k$. In other terms,

[TABLE]

holds isometrically, and $h_{\log n}=z^{\alpha}$ whenever $n=\mathfrak{p}^{\alpha}$.

There is a useful reformulation of the Dirichlet group $(\mathbb{T}^{\infty},\beta_{\mathbb{T}^{\infty}})$. Denote by $\Xi$ the set of all characters $\chi:\mathbb{N}\to\mathbb{T}$, i.e. $\chi$ is completely multiplicative in the sense that $\chi(nm)=\chi(n)\chi(m)$ for all $n,m$. So every character is uniquely determined by its values on the primes. If we on $\Xi$ consider pointwise multiplication, then

[TABLE]

is a group isomorphism which turns $\Xi$ into a compact abelian group. The Haar measure $d\chi$ on $\Xi$ is the push forward of the normalized Lebesgue measure $dz$ on $\mathbb{T}^{\infty}$ through $\iota^{-1}$. Hence also $\Xi$ together with

[TABLE]

forms a $(\log n)$-Dirichlet group.

Recall from [4, Lemma 3.11] that, given a Dirichlet group $(G,\beta)$ and $f\in L_{1}(G)$, for almost all $\omega\in G$ there are locally Lebesgue integrable functions $f_{\omega}\colon\mathbb{R}\to\mathbb{C}$ such that $f_{\omega}(t)=f(\omega\beta(t))$ almost everywhere on $\mathbb{R}$. As we will see later, this way to ’restrict’ functions on the group $G$ to $\mathbb{R}$, establishes a sort of bridge between Fourier analysis on a $\lambda$-Dirichlet group $G$ and Fourier analysis on $\mathbb{R}$.

In this context, the classical Poisson kernel $P_{u}(t)=\frac{1}{\pi}\frac{u}{u^{2}+t^{2}}\colon\mathbb{R}\to\mathbb{R}$, where $u>0$, plays a crucial role. Since $P_{u}\in L_{1}(\mathbb{R})$ has norm $1$, its push forward under $\beta$ leads to a regular Borel probability measure $p_{u}$ on $G$. We call $p_{u}$ the Poisson measure on $G$, and note that $\widehat{p_{u}}(h_{x})=e^{-u|x|}$ for all $x\in\widehat{\beta}(\widehat{G})$.

1.2. The reflexive case – functions

Given a compact and abelian group $G$ with Haar measure $m$ and a class $\mathcal{F}$ of functions in $L_{p}(G),1\leq p\leq\infty$, one of the fundamental questions of Fourier analysis certainly is to ask for necessary and sufficient conditions on $\mathcal{F}$ under which the Fourier series $\sum_{\gamma\in\widehat{G}}\widehat{f}(\gamma)\gamma$ of each $f\in\mathcal{F}$ approximates $f$ in a reasonable way – e.g. almost everywhere pointwise or in the $L_{p}$-norm, and with respect to a reasonable summation method like ordinary or Cesáro summation.

In the following we will carefully distinguish the reflexive case $1<p<\infty$ from the non-reflexive cases $p=1$ and $\infty$.

Theorem 1.1**.**

Let $(G,\beta)$ be a $\lambda$-Dirichlet group and $1<p<\infty$. There is a constant $\text{CH}_{p}>0$ such that for every $f\in H_{p}^{\lambda}(G)$ we have

[TABLE]

In particular, $\sum\widehat{f}(h_{\lambda_{n}})h_{\lambda_{n}}$ approximates $f$ almost everywhere pointwise and in the $H_{p}$-norm.

For $\lambda=(n)$ and the Dirichlet group $(\mathbb{T},\beta_{\mathbb{T}})$ this is the celebrated Carleson-Hunt theorem, and the constant $\text{CH}_{p}$ mentioned above in fact equals the one from the maximal inequality in the CH-theorem for one variable. Based on this one variable case and a method from [8], Hedenmalm-Saksman in [12, Theorem 1.5] extend the CH-theorem to functions $f\in H_{2}(\mathbb{T}^{\infty})$, which in the preceding theorem is reflected by the $(\log n)$-Dirichlet group $(\mathbb{T}^{\infty},\beta_{\mathbb{T}^{\infty}})$. The general case given above has to be credited to Duy from [6]; for our reformulation within the setting of arbitrary $\lambda$-Dirichlet groups we refer to [5].

The CH-theorem in one variable fails for $p=1$, so clearly the preceding extension fails in this case. On the other hand, it is well-known that every function $f\in H_{1}(\mathbb{T})$ almost everywhere equals the pointwise limit of its Cesàro means (see e.g. [9, Theorem 3.4.4, p.207]), i.e.

[TABLE]

for almost all $z\in\mathbb{T}$. Moreover, this is also true if the limits are taken with respect to the $H_{1}$-norm. So it seems natural to consider for a given $f\in H_{1}(\mathbb{T}^{\infty})$ the Cesàro means of the partial sums

[TABLE]

and to ask whether almost everywhere pointwise and/or in the $H_{1}$-norm

[TABLE]

We will later see that this is in general false – but true, if we change Cesàro summation by more adapted summation methods invented in [10] by Hardy and M. Riesz within the setting of general Dirichlet series.

1.3. Riesz means – functions

The following definitions are inspired by [10]. Let $\lambda$ be a frequency, $k\geq 0$, and $\sum c_{n}$ a series in a Banach space $X$. Then we call the series $\sum c_{n}$ $(\lambda,k)$-Riesz summable if the limit

[TABLE]

exists, and we call the finite sums

[TABLE]

first $(\lambda,k)$-Riesz means of $\sum c_{n}$ of length $x>0$.

Hardy and Riesz in [10] isolated the following fundamental properties of Riesz summability; the results are (in the order of the proposition) taken from [10, Theorem 16, p. 29, Theorem 17, p. 30, and Theorem 21, p. 36].

Proposition 1.2**.**

Let $\lambda$ be a frequency, $k\geq 0$, and $\sum c_{n}$ a series in a Banach space $X$.

• (1)

First theorem of consistency: If $\sum c_{n}$ is $(\lambda,k)$-Riesz summable, then it is $(\lambda,\ell)$-Riesz summable for any $k\leq\ell$, and the associated limits coincide. In particular, if $\sum c_{n}$ is summable (i.e. the series converges), then for all $k>0$

[TABLE]

• (2)

Second theorem of consistency: If $\sum c_{n}$ is $(e^{\lambda},k)$-Riesz summable, then $\sum c_{n}$ is $(\lambda,k)$-Riesz summable, and the associated limits coincide.

• (3)

If $\sum c_{n}$ is $(\lambda,k)$-summable summable with limit $C$, then

[TABLE]

Note that, if $\sum c_{n}$ in view of Proposition 1.2, (2) is not only $(\lambda,k)$-Riesz summable but even $(e^{\lambda},k)$-Riesz summable, then

[TABLE]

we refer to the finite sums

[TABLE]

as the second $(\lambda,k)$-Riesz means of $\sum c_{n}$ of length $x>0$.

Take now some $\lambda$-Dirichlet group $(G,\beta)$ and $f\in H_{1}^{\lambda}(G)$. Then we call the Fourier series of $f$ $(\lambda,k)$-Riesz summable in $\omega\in G$ if it is $(\lambda,k)$-Riesz summable in $\omega\in G$, in other terms the limit

[TABLE]

exists. It is then needless to say what is meant by the phrase ’the Fourier series of $f$ is $(\lambda,k)$-Riesz summable in the $H_{p}$-norm’. Moreover, the polynomial

[TABLE]

is the so-called first $(\lambda,k)$-Riesz mean of $f$ of length $x>0$ , and

[TABLE]

the second $(\lambda,k)$-Riesz mean of $f$ of length $x>0$. Observe that if the Fourier series of $f$ is $(e^{\lambda},k)$-Riesz summable in $\omega\in G$, then as in (5)

[TABLE]

Let us again for a moment concentrate on the two in a sense extrem frequencies $\lambda=(n)$ and $\lambda=(\log n)$.

As mentioned $\lambda=(n)$ together with $(\mathbb{T},\beta_{\mathbb{T}})$ forms a $\lambda$-Dirichlet group. Then the $(\lambda,1)$-Riesz mean of $f\in H_{1}(\mathbb{T})=\mathcal{H}_{1}^{\lambda}(\mathbb{T})$ of length $x$ equals

[TABLE]

which for $x=N\in\mathbb{N}$ is nothing else than the Cesàro mean of the $N$th partial sum of the Fourier series of $f$ considered in (3).

In this sense Riesz means generalize the Cesàro means for functions on $\mathbb{T}$ to the much wider setting of functions on Dirichlet groups.

Let us also consider $\lambda=(\log n)$ and the $\lambda$-Dirichlet group $(\mathbb{T}^{\infty},\beta_{\mathbb{T}^{\infty}})$. For $f\in H_{1}(\mathbb{T}^{\infty})=H_{1}^{\lambda}(\mathbb{T}^{\infty})$ we refer to the first $(\lambda,k)$-Riesz means of $f$, that are

[TABLE]

as the logarithmic means of $f$. Observe also, that in this case

[TABLE]

and hence for $N\in\mathbb{N}$ and $k=1$

[TABLE]

Remark 1.3**.**

Let $f\in H^{\lambda}_{1}(G)$, $\omega\in G$, and $k\geq 0$. Then for $\lambda=(n)$ the following are equivalent:

• (1)

The Fourier series of $f$ converges at $\omega$.

• (2)

The Fourier series of $f$ is $(e^{\lambda},k)$-Riesz summable at $\omega$.

In this case the limits coincide, and a similar result holds true, whenever we replace convergence in $\omega$ by convergence with respect to the $H_{p}$-norm, $1\leq p\leq\infty$.

Proof.

Part (3) of Proposition 1.2 proves the implication (2) $\Rightarrow$ (1), and the reversed direction follows from Proposition 1.2 (1). ∎

So for the frequency $\lambda=(n)$, Riesz summability by second means seems not to be particularly interesting.

After all this, let us finally indicate the main challenge of this article: For which frequencies $\lambda$ and which $\lambda$-Dirichlet groups $(G,\beta)$ do we for all $f\in H_{1}^{\lambda}(G)$ have

[TABLE]

almost everywhere on $G$ and/or in the $H_{1}$-norm?

1.4. Hardy spaces of Dirichlet series

Given a frequency $\lambda=(\lambda_{n})$, a $\lambda$-Dirichlet series is a (formal) sum of the form $D=\sum a_{n}e^{-\lambda_{n}s}$, where $s$ is a complex variable and the sequence $(a_{n})$ form the so-called Dirichlet coefficients of $D$. Finite sums of the form $D=\sum_{n=1}^{N}a_{n}e^{-\lambda_{n}s}$ we call Dirichlet polynomials. By $\mathcal{D}(\lambda)$ we denote the space of all $\lambda$-Dirichlet series. It is well-known that if $D=\sum a_{n}e^{-\lambda_{n}s}$ converges in $s_{0}\in\mathbb{C}$, then it also converges for all $s\in\mathbb{C}$ with $Res>Res_{0}$, and its limit function $f(s)=\sum_{n=1}^{\infty}a_{n}e^{-\lambda_{n}s}$ defines a holomorphic function on $[Re>\sigma_{c}(D)]$, where

[TABLE]

determines the so-called abscissa of convergence.

In [4] we introduce an $\mathcal{H}_{p}$-theory of general Dirichlet series extending Bayart’s $\mathcal{H}_{p}$-theory of ordinary Dirichlet series $\sum a_{n}n^{-s}$ (where $\lambda=(\log n)$) from [1] (see also e.g. [3], [14], and [20] for more information on the ’ordinary’ case). Let us briefly recall the general framework from [4], which in particular shows, that there are several ways to produce Dirichlet series.

Fixing a $\lambda$-Dirichlet group $(G,\beta)$, we define $\mathcal{H}_{p}(\lambda)$, where $1\leq p\leq\infty$, to be the space of all (formal) $\lambda$-Dirichlet series $D=\sum a_{n}e^{-\lambda_{n}s}$ for which there is $f\in H_{p}^{\lambda}(G)$ such that that $a_{n}=\widehat{f}(h_{\lambda_{n}})$ for all $n\in\mathbb{N}$. Endowed with the norm $\|D\|_{p}:=\|f\|_{p}$, we obtain a Banach space.

Note that by (1), the definition of $\mathcal{H}_{p}(\lambda)$ is independent of the chosen $\lambda$-Dirichlet group, and we by definition obtain the onto isometry

[TABLE]

for historical reasons we call this mapping Bohr transform. From [4, Theorem 3.26] recall the following internal description of $\mathcal{H}_{p}(\lambda)$: Since $D=\sum_{n=1}^{N}a_{n}e^{-\lambda_{n}s}$, considered as a function on the imaginary line, defines an almost periodic function, the limit

[TABLE]

exists and defines a norm on the space $Pol(\lambda)$ of all $\lambda$-Dirichlet polynomials. Then $\mathcal{H}_{p}(\lambda)$ is the completion of \big{(}Pol(\lambda),\|\cdot\|_{p}\big{)}.

1.5. Transference

We want to understand, how (pointwise) summability properties of the Fourier series $\sum\widehat{f}(h_{\lambda_{n}})h_{\lambda_{n}}$ of functions $f\in H_{1}^{\lambda}(G)$ transfer to summability properties of their associated Dirichlet series $D:=\mathcal{B}(f)$, and vice versa. Slightly more precise, but still vague, we try to figure out how summation of these Fourier series by first or second Riesz means influences the convergence properties of so-called vertical limits of $D$.

Given a $\lambda$-Dirichlet series $D=\sum a_{n}e^{-\lambda_{n}s}$ and $z\in\mathbb{C}$, we call the Dirichlet series

[TABLE]

the translation of $D$ about $z$, and we distinguish between horizontal translations $D_{u},u\in\mathbb{R}$, and vertical translations $D_{i\tau},\tau\in\mathbb{R}$.

If $(G,\beta)$ is a $\lambda$-Dirichlet group and $D\in\mathcal{H}_{p}(\lambda)$ is associated to $f\in H_{p}^{\lambda}(G)$, then for each $u>0$ the horizontal translation $D_{u}$ corresponds to the convolution of $f$ with the Poisson measure $p_{u}$, i.e. $\mathcal{B}(f*p_{u})=D_{u}$ (compare coefficients). In particular, we have that $D_{u}\in\mathcal{H}_{p}(\lambda)$.

Moreover, each Dirichlet series of the form

[TABLE]

is said to be a vertical limit of $D$. Examples are vertical translations $D_{i\tau}$ with $\tau\in\mathbb{R}$, and the terminology is explained by the fact that each vertical limit may be approximated by vertical translates. More precisely, given $D=\sum a_{n}e^{-\lambda_{n}s}$ which converges absolutely on the right half-plane, for every $\omega\in G$ there is a sequence $(\tau_{k})_{k}\subset\mathbb{R}$ such that $D_{i\tau_{k}}$ converges to $D^{\omega}$ uniformly on $[Re>\varepsilon]$ for all $\varepsilon>0$. Assume conversely that for $(\tau_{k})_{k}\subset\mathbb{R}$ the vertical translations $D_{i\tau_{k}}$ converge uniformly on $[Re>\varepsilon]$ for every $\varepsilon>0$ to a holomorphic function $f$ on $[Re>0]$. Then there is $\omega\in G$ such that $f(s)=\sum_{n=1}^{\infty}a_{n}\omega(h_{\lambda_{n}})e^{-\lambda_{n}s}$ for all $s\in[Re>0]$ . For all this see [4, Proposition 4.6].

The following lemma (to be proved in Section 3.4) is our ’bridge’ comparing almost everywhere Riesz-summability of the Fourier series of a function $f\in H_{1}^{\lambda}(G)$ with the convergence of almost all vertical limits $D^{\omega}$ of its associated Dirichlet series $D=\mathfrak{B}(f)$ almost everywhere on the imaginary axis.

Lemma 1.4**.**

Let $(G,\beta)$ be a $\lambda$-Dirichlet group, and $f_{n},f\in H_{1}^{\lambda}(G)$. Then the following are equivalent:

• (1)

$\lim_{n\to\infty}f_{n}(\omega)=f(\omega)$*     for almost all $\omega\in G$*

• (2)

$\lim_{n\to\infty}(f_{n})_{\omega}(t)=f_{\omega}(t)$*     for almost all $\omega\in G$ and for almost all $t\in\mathbb{R}$.*

In particular, if all $f_{n}$ are polynomials and $D_{n}\in\mathcal{H}_{1}(\lambda)$ are the Dirichlet polynomials associated to $f_{n}$ under the Bohr transform, then $(1)$ and $(2)$ are equivalent to each of the following two further statements:

• (3)

$\lim_{n\to\infty}D^{\omega}_{n}(0)=f(\omega)$*     for almost all $\omega\in G$*

• (4)

$\lim_{n\to\infty}D^{\omega}_{n}(it)=f_{\omega}(t)$*     for almost all $\omega\in G$ and for almost all $t\in\mathbb{R}$ .*

Note that the question we formulated in (11) then reads: For which frequencies $\lambda$ and for which $\lambda$-Dirichlet groups $(G,\beta)$ is it true that for every $D\in\mathcal{H}_{1}(\lambda)$ we for almost all $\omega\in G$ have

[TABLE]

almost everywhere on the imaginary axis?

1.6. The reflexive case – Dirichlet series

The following result is an immediate consequence of Theorem 1.1 and Lemma 1.4.

Theorem 1.5**.**

Let $\lambda$ be a frequency, $(G,\beta)$ a $\lambda$-Dirichlet group, and $1<p<\infty$. Then for every $D=\sum a_{n}e^{-\lambda_{n}s}\in\mathcal{H}_{p}(\lambda)$

[TABLE]

In particular, almost all vertical limits $D^{\omega}$ converge almost everywhere on the imaginary axis, and consequently also on the right half-plane.

For $p=2$ Helson in [13, §2] proves convergence on $[Re>0]$ under Bohr’s condition for $\lambda$, i.e.

[TABLE]

(see also [14, Theorem 9, p. 29]), and in the ordinary case and $1\leq p<\infty$ this is done by Bayart [1]. Still in the ordinary case, convergence on the imaginary axis $[Re=0]$ for $p=2$ has to be credited to Hedenmalm-Saksman [12], and for the full scale $1<p<\infty$ this is observed in [5].

But for $p=1$ the first two assertions of the preceding result are false. Otherwise by Lemma 1.4 we would see that all Fourier series of functions $f\in H_{1}(\mathbb{T}^{\infty})$ converge pointwise almost everywhere which we know is false (even in the one variable case). The third statement we only know under the additional Landau condition $(LC)$ for $\lambda$ (see again [5]), i.e.

[TABLE]

this in fact is a condition weaker than $(BC)$.

Theorem 1.6**.**

Let $\lambda$ be a frequency with $(LC)$ and $(G,\beta)$ a $\lambda$-Dirichlet group. Then for $D=\sum a_{n}e^{-\lambda_{n}s}\in\mathcal{H}_{1}(\lambda)$ almost all vertical limits $D^{\omega}$ of $D$ converge on $[\text{Re}>0]$. More precisely, if $f\in H_{1}^{\lambda}(G)$ is associated to $D$, then there is a null set $N\subset G$ such that for $\omega\notin N$, all $u>0$ and almost all $t\in\mathbb{R}$

[TABLE]

See also Section 2.2, where we show that in the ordinary case Theorem 1.5 is false for $p=1$, even if we there replace summation of the series by the weaker Cesàro summation. Alternative more adapted summation methods have to be taken into account which we describe now (recall also the discussion from (4)).

1.7. Riesz means – Dirichlet series

Here we repeat some fundamental definitions and results on Riesz-summability of general Dirichlet series from [10]. Fix some frequency $\lambda$, some $k\geq 0$, and a $\lambda$-Dirichlet series $D=\sum a_{n}e^{-\lambda_{n}s}$. The first $(\lambda,k)$-Riesz mean of $D$ of length $x>0$ is given by the Dirichlet polynomial

[TABLE]

We say that a Dirichlet series $D=\sum a_{n}e^{-\lambda_{n}s}$ is $(\lambda,k)$-Riesz summable at $s_{0}\in\mathbb{C}$, if the limit

[TABLE]

exists, and $D=\sum a_{n}e^{-\lambda_{n}s}$ is $(\lambda,k)$-Riesz summable in $\mathcal{H}_{p}(\lambda)$ whenever this limit exists in $\mathcal{H}_{p}(\lambda)$. As in (5), if $D$ is (even) $(e^{\lambda},k)$-Riesz summable, then we have

[TABLE]

where

[TABLE]

is what we call the second $(\lambda,k)$-Riesz mean of $D$ of length $x>0$.

Let us again comment on the ordinary case $\lambda=(\log n)$. Then the first $(\lambda,k)$-Riesz mean of $D=\sum a_{n}n^{-s}$ of length $x$ is given by

[TABLE]

Hardy and Riesz in [10] call it the logarithmic mean of $D$ of length $x$. As in (10), we for $\lambda=(\log n)$ and $x=N\in\mathbb{N}$ obtain Cesàro means,

[TABLE]

From Remark 1.3 we may deduce the following equivalence for the case $\lambda=(n)$.

Remark 1.7**.**

Let $D=\sum a_{n}e^{-\lambda_{n}s}$, $s_{0}\in\mathbb{C}$, and $k\geq 0$. Then for $\lambda=(n)$ the following are equivalent:

• (1)

$D$* converges at $s_{0}$*

• (2)

$D$* is $(e^{\lambda},k)$-summable at $s_{0}$*

Moreover, the limits coincide, and the analog result holds true, whenever we replace convergence in $s_{0}$ by convergence with respect to the $\mathcal{H}_{p}$-norm, $1\leq p\leq\infty$.

All results collected in Proposition 1.2 may be translated to $\lambda$-Dirichlet series. We do this in terms of Riesz-abscissas of convergence. Define for $D=\sum a_{n}e^{-\lambda_{n}s}$

[TABLE]

Then it is proved in [10, Theorem 24 and 25, p. 43] that $D$ converges on the half-plane $[Re>\sigma_{c}^{\lambda,k}(D)]$ whereas it diverges on $[Re<\sigma_{c}^{\lambda,k}(D)]$. Moreover, the limit function is holomorphic (see [10, Theorem 27, p. 44]). Obviously, we have that

[TABLE]

From Hardy and Riesz [10, Theorem 16, p. 29 and Theorem 30, p. 45] (see also again Proposition 1.2) we know the following.

Proposition 1.8**.**

Let $\lambda$ be a frequency and $0\leq k<\ell$. Then for every $\lambda$-Dirichlet series $D$ we have

• (1)

$\sigma_{c}^{\lambda,k}(D)\leq\sigma_{c}^{\lambda,\ell}(D)$**

• (2)

$\sigma_{c}^{e^{\lambda},k}(D)=\sigma_{c}^{\lambda,k}(D)$**

Finally, we note that for $f\in H_{1}^{\lambda}(G)$, where $(G,\beta)$ is some $\lambda$-Dirichlet group and $D\in\mathcal{H}_{1}(\lambda)$ with $D:=\mathcal{B}(f)$, we for all $k,x>0$ have

[TABLE]

2. Results

We start summarizing our results according to the following table of content, and then later on all proofs will be given in the final Section 3.

• •

Pointwise approximation by first Riesz means (Section 2.1)

• •

Failure of pointwise approximation by second Riesz means (Section 2.2)

• •

A Hardy-Littlewood maximal operator (Section 2.3)

• •

Norm approximation by Riesz means (Section 2.4)

• •

Uniformly almost periodic functions (Section 2.5)

2.1. Pointwise approximation by first Riesz means

Generally speaking, in order to obtain almost everywhere convergence of the Riesz means of some $f\in H^{\lambda}_{1}(G)$, it is sufficient to prove an adequate maximal inequality.

For that purpose recall that for some measure space $(\Omega,\mu)$ the weak $L_{1}$-space $L_{1,\infty}(\mu)$ is given by all measurable functions $f\colon\Omega\to\mathbb{C}$ for which there is a constant $C>0$ such that for all $\alpha>0$ we have

[TABLE]

Together with $\|f\|_{1,\infty}:=\inf C$ the space $L_{1,\infty}(\mu)$ becomes a quasi Banach space (see e.g. [9, §1.1.1 and §1.4]), where the triangle inequality holds with constant $2$.

The following maximal inequality forms the core of this article.

Theorem 2.1**.**

Let $\lambda$ be a frequency, $k>0$ and $(G,\beta)$ a $\lambda$-Dirichlet group. Then

[TABLE]

defines a bounded sublinear operator from $H_{1}^{\lambda}(G)$ to $L_{1,\infty}(G)$, and from $H_{p}^{\lambda}(G)$ to $L_{p}(G)$, whenever $1<p\leq\infty$.

By a standard argument (to be formalized in Lemma 3.6), we deduce from Theorem 2.1 for each $f\in H_{1}^{\lambda}(G)$ almost everywhere summability by first Riesz means of the Fourier series $\sum\widehat{f}(h_{\lambda_{n}})h_{\lambda_{n}}$, but also of the translated Fourier series $\sum\widehat{f}(h_{\lambda_{n}})e^{-u\lambda_{n}}h_{\lambda_{n}}$.

Corollary 2.2**.**

Let $f\in H_{1}^{\lambda}(G)$ and $k>0$. Then for almost all $\omega\in G$ we have

[TABLE]

Moreover, there is a null set $N\subset G$ such that for all $u>0$ and all $\omega\notin N$

[TABLE]

Note that for some fixed $u>0$ the result from (20) is immediate from (19) since $f*p_{u}\in H_{1}^{\lambda}(G)$, but the point here is that the null set $N$ in fact can be chosen to be independent of $u$.

In particular, (19) reproves (3) with the choice $\lambda=(n)$, $k=1$ and the Dirichlet group $(\mathbb{T},\beta_{\mathbb{T}})$. Since first Riesz means of order $k=0$ are precisely partial sums, Corollary 2.2 may fail for $k=0$ (like it does for $\lambda=(n)$). Let us also revisit the ordinary case $\lambda=(\log n)$. Then Theorem 2.1 gives the following substitute for the failure of (4).

Corollary 2.3**.**

Let $f\in H_{1}(\mathbb{T}^{\infty})$ and $k>0$. Then for almost all $z\in\mathbb{T}^{\infty}$

[TABLE]

that is, $f$ almost everywhere is the pointwise limit of its logarithmic means.

Let us explain in which sense (19) is the limit case of (20).

Proposition 2.4**.**

Let $\lambda$ be arbitrary. Then the operator

[TABLE]

defines a bounded sublinear operator from $L_{1}(G)$ to $L_{1,\infty}(G)$, and from $L_{p}(G)$ to $L_{p}(G)$, provided $1<p\leq\infty$. In particular, for almost all $\omega\in G$

[TABLE]

Obviously, Proposition 2.4 implies that for almost all $\omega\in G$

[TABLE]

explaining, why (19) is the limit case of (20).

For the $(n)$-Dirichlet group $(\mathbb{T},\beta_{\mathbb{T}})$ all this is linked with Fatou’s famous theorem on radial limits. Observe that, if $f\in H_{1}(\mathbb{T})$ and $0<r<1$, then with the choice $u:=-\log(r)$ we have

[TABLE]

and Proposition 2.4 implies, that for almost all $z\in\mathbb{T}$

[TABLE]

In the terminology of [16, Theorem 11.23] this says that for every $f\in H_{p}(\mathbb{T})$ the so-called Poisson integral $P[f](w):=\sum_{n=0}^{\infty}\widehat{f}(n)w^{n},w\in\mathbb{D}$ has radial limits almost everywhere on $\mathbb{D}$ . This is a crucial step of the proof of Fatou’s theorem, which states, that if $f\in H_{p}(\mathbb{D})$, $1\leq p\leq\infty$, then for almost all $z\in\mathbb{T}$ the radial limits $g^{*}(z):=\lim_{r\to 1}g(rz)$ exist and define a function from $H_{p}(\mathbb{T})$. So in this sense, Proposition 2.4 extends (22) from $H_{p}^{(n)}(\mathbb{T})$ to $H_{p}^{\lambda}(G)$.

Let us again revisit Theorem 2.1. If $p=1$, then the function $R_{\max}^{\lambda,k}(f)$ for $f\in H^{\lambda}_{1}(G)$ may fail to be in $L_{1}(G)$, like it does for $\lambda=(n)$ and $(\mathbb{T},\beta_{\mathbb{T}})$. On the other hand, philosophically speaking, for $f\in H^{\lambda}_{1}(G)$ a horizontal translation of $f$ by $u>0$, that is

[TABLE]

improves $f$ considerably. Then, as shown in the following result, $R_{\max}^{\lambda,k}(f_{u})$ for any $u,k>0$ indeed belongs to $L_{1}(G)$. Of course we we already know from Theorem 2.1 that $R_{\max}^{\lambda,k}(f_{u})\in L_{p}(G)$ whenever $f\in H_{p}^{\lambda}(G)$ and $1<p\leq\infty$.

Theorem 2.5**.**

Let $u,k>0$. Then there is a constant $C=C(u,k)$ such that for all $\lambda$, $1\leq p<\infty$ and $f\in H_{p}^{\lambda}(G)$ we have

[TABLE]

Compared with Theorem 2.1 the relevant part of this result is the case $p=1$. But also the case of arbitrary $p$’s seems interesting since in the above inequality the constant $C=C(u,k)$ does not depend on $p$.

Finally, we discuss Riesz summation of Dirichlet series. We use the ’bridge’ from Lemma 1.4 to rephrase Corollary 2.2 in terms of the summability of almost all vertical limits of $\mathcal{H}_{1}$-Dirichlet series by first Riesz means almost everywhere on the imaginary axis.

Corollary 2.6**.**

Let $\lambda$ be a frequency, $k>0$, $f\in H_{1}^{\lambda}(G)$, and $D$ its associated Dirichlet series in $\mathcal{H}_{1}(\lambda)$. Then there is a null set $N\subset G$ such that for all $\omega\notin N$

[TABLE]

[TABLE]

and

[TABLE]

Note that by Theorem 1.6 the case $k=0$ in (25) is known, if $\lambda$ satisfies $(LC)$.

In view of (13) the following maximal inequality may be considered as an internal variant of Theorem 2.5.

Theorem 2.7**.**

Let $u,k>0$. Then there is a constant $C=C(u,k)$ such that for all $\lambda$, $1\leq p<\infty$, and $D=\sum a_{n}e^{-\lambda_{n}s}\in\mathcal{H}_{p}(\lambda)$ we for almost all $\omega\in G$ have

[TABLE]

All proofs of this section are given in the Sections 3.2, 3.3, 3.4, and 3.5.

2.2. Failure of pointwise approximation by second Riesz means

Let $\lambda$ be a frequency, $k>0$, and $(G,\beta)$ a $\lambda$-Dirichlet group. From Corollary 2.2 we know that the Fourier series of each $f\in H_{1}^{\lambda}(G)$ is $(\lambda,k)$-Riesz summable with limit $f$, i.e. $f$ is almost everywhere approximable by its first Riesz means $R^{\lambda,k}_{x}(f)$. Let us discuss whether in view of Proposition 1.2, (2) the Fourier series of each $f\in H_{1}^{\lambda}(G)$ is even $(e^{\lambda},k)$-Riesz summable with limit $f$, i.e. $f$ is almost everywhere approximable by its second Riesz means $S^{\lambda,k}_{x}(f)$ (see (6)).

For $\lambda=(n)$ and the $\lambda$-Dirichlet group $(\mathbb{T},\beta_{\mathbb{T}})$ we know that the Fourier series of each $f\in H_{1}^{\lambda}(\mathbb{T})=H_{1}^{\lambda}(\mathbb{T})$ is Cesáro summable almost everywhere with limit $f$, i.e. it is $(\lambda,1)$-Riesz summable with limit $f$ (as mentioned this is a particular case of Corollary 2.2). But it is definitely not $(e^{\lambda},1)$-Riesz summable since otherwise by Proposition 1.3 each $f\in H_{1}(\mathbb{T})$ would have an almost everywhere convergent Fourier series.

On the other hand look at $\lambda=(\log n)$ and the $\lambda$-Dirichlet group $(\mathbb{T}^{\infty},\beta_{\mathbb{T}^{\infty}})$. We have already mentioned Corollary 2.3 which shows that the Fourier series of every $f\in H_{1}(\mathbb{T}^{\infty})=H_{1}^{\lambda}(\mathbb{T}^{\infty})$ is $(\lambda,1)$-Riesz summable with limit $f$. Assume that the Fourier series of each such $f$ may even be almost everywhere $(e^{\lambda},1)$-summable, i.e. it is almost everywhere on $\mathbb{T}^{\infty}$ approximable by its second Riesz means:

[TABLE]

Then for every $f\in H_{1}(\mathbb{T})\subset H_{1}(\mathbb{T}^{\infty})$ almost everywhere on $\mathbb{T}$

[TABLE]

In other words the Fourier series of every $f\in H_{1}(\mathbb{T})$ is almost everywhere $((2^{j}),1)$-Riesz summable with limit $f$. By Proposition 1.2, this implies that each such Fourier series is almost everywhere summable on $\mathbb{T}$, a contradiction.

We collect the preceding information in the following

Remark 2.8**.**

Given a frequency $\lambda$ and a $\lambda$-Dirichlet group $(G,\beta)$, it is in general not true, that each $f\in H_{1}^{\lambda}(G)$ has an almost everywhere $(e^{\lambda},1)$-summable Fourier series. Counterexamples are $\lambda=(n)$ and $(\mathbb{T},\beta_{\mathbb{T}})$, as well as $\lambda=(\log n)$ and $(\mathbb{T}^{\infty},\beta_{\mathbb{T}^{\infty}})$.

Corollary 2.6 shows that, given a $\lambda$-Dirichlet group $(G,\beta)$ and $k>0$, almost all vertical limits $D^{\omega}$ of a Dirichlet series $D\in\mathcal{H}_{1}(\lambda)$ are $(\lambda,k)$-Riesz summable almost everywhere on the vertical line. Are they even $(e^{\lambda},k)$-Riesz summable? Remark 2.8 and Lemma 1.4 show that the answer in general in no.

But something can be saved. By Corollary 2.6 and Proposition 1.8 almost all vertical limits of Dirichlet series $D\in\mathcal{H}_{1}(\lambda)$ are $(\lambda,k)$-Riesz summable on $[Re>0]$, and so $(e^{\lambda},k)$-Riesz summable on $[Re>0]$. As a consequence of Theorem 2.5 we are going to deduce the following quantitative version of this fact.

Theorem 2.9**.**

Let $u,k>0$. Then there is a constant $C=C(u,k)$ such that for all $\lambda$, $1\leq p<\infty$ and $f\in H_{p}^{\lambda}(G)$ we have

[TABLE]

One could be tempted to believe that this maximal inequality is an immediate consequence of Theorem 2.5 (applying it to $e^{\lambda}$ instead of $\lambda$). But this is not true since we here consider functions in $H_{p}^{\lambda}(G)$ and not $H_{p}^{e^{\lambda}}(G)$. The proof is given in Section 3.5.

2.3. A Hardy-Littlewood maximal operator

One of the central tools needed for the proof of Theorem 2.1 is given a Hardy-Littlewood maximal operator adapted to Fourier analysis on Dirichlet groups. If $f\in L_{1}(G)$, where $(G,\beta)$ is any Dirichlet group, then we define

[TABLE]

here $I$ stands for any interval in $\mathbb{R}$ and $|I|$ for its Lebesgue measure. Recall that $f_{\omega}$ for almost all $\omega\in G$ is a locally integrable function on $\mathbb{R}$ with $f_{\omega}(t)=f(\omega\beta(t))$ for almost all $t\in\mathbb{R}$, and so $\overline{M}(f)(\omega)$ is defined almost everywhere.

Note that the definition of $\overline{M}$ actually depends on the choice of the Dirichlet group $(G,\beta)$, although for simplicity our notation does not indicate this. Moreover observe that (26) for $(\mathbb{T},\beta_{\mathbb{T}})$ precisely gives the classical Hardy-Littlewood maximal operator on $\mathbb{T}$ (see e.g. [16, (3), p. 241]).

Theorem 2.10**.**

Let $(G,\beta)$ be a Dirichlet group. Then the sublinear operator $\overline{M}$ is bounded from $L_{1}(G)$ to $L_{1,\infty}(G)$ and from $L_{p}(G)$ to $L_{p}(G)$, whenever $1<p\leq\infty.$

In Section 3.2 we see that the proof of Theorem 2.1 reduces to Theorem 2.10. The following corollary is a consequence of independent interest.

Corollary 2.11**.**

Let $f\in L_{1}(G)$. Then for almost all $\omega\in G$ we have

[TABLE]

and

[TABLE]

Note that (27) may be interpret as a ’differentiation theorem’ for integrable functions on Dirichlet groups.

Given $1\leq p<\infty$, recall that the so-called Marcinkiewicz space $\mathcal{M}_{p}(\mathbb{R})$ of all $f\in L_{loc}^{p}(\mathbb{R})$ for which

[TABLE]

exists, contains all trigonometric polynomials of the form $q(t):=\sum_{n=1}^{N}a_{n}e^{-itx_{n}}$ with $x_{n}\in\mathbb{R}$ for $1\leq n\leq N$, and in this case we have that

[TABLE]

Then the Besicovitch space $\mathcal{B}_{p}(\mathbb{R})$ is defined to be the closure of the trigonometric polynomials in $(\mathcal{M}_{p}(\mathbb{R}),\|\cdot\|_{p})$. By density, we additionally have $\mathcal{B}_{p}(\mathbb{R})=L_{p}(\overline{\mathbb{R}})$. Moreover, from [4] we know that $L_{p}(G)\subset L_{p}(\overline{\mathbb{R}})$ (isometrically) for all Dirichlet groups $(G,\beta)$, and so $L_{p}(G)\subset\mathcal{B}_{p}(\mathbb{R})$. Now given $f\in L_{p}(G)$, it seems difficult to determine the corresponding function in $\mathcal{B}_{p}(\mathbb{R})$. But we are going to deduce from $\eqref{besiconorm2}$ that at least for almost all $\omega\in G$ the corresponding function for the translations $f(\omega\cdot)$ is given by $f_{\omega}$.

Remark 2.12**.**

Given $f\in L_{p}(G)$, then $f_{\omega}\in\mathcal{B}_{p}(\mathbb{R})$ for almost all $\omega\in G$ and

[TABLE]

All proofs of this section are given in Section 3.1 and Section 3.3.

2.4. Norm approximation by first and second Riesz means

From [4] we know that for any frequency $\lambda$ the sequence $(h_{\lambda_{n}})$ forms a Schauder basis for $H_{p}^{\lambda}(G)$, provided $1<p<\infty$. Note that this can also be seen as a straight forward consequence of the maximal inequality from Theorem 1.5.

Consequently for $1<p<\infty$ every function from $H_{p}^{\lambda}(G)$ is approximated by its first and second Riesz means with respect to the norm. As shown by the next result, this for $p=1$ and the first Riesz means is still true.

Theorem 2.13**.**

Let $f\in H_{1}^{\lambda}(G)$. Then for all $k>0$

[TABLE]

the limit taken with respect to the $H_{1}$-norm.

But for the frequencies $\lambda=(n)$ and $\lambda=(\log n)$ this result in general fails if we replace first Riesz means by second Riesz means. This is shown exactly as in Remark 2.8 replacing almost everywhere convergence by convergence in the norm.

Remark 2.14**.**

For $\lambda=(n)$ and $\lambda=(\log n)$ the Fourier series of some $f\in H_{1}^{\lambda}(G)$ in general is not $(e^{\lambda},1)$-Riesz summable in the $H_{p}$-norm, i.e. the second $(\lambda,1)$-Riesz means do not approximate the function in the $H_{p}$-norm.

The proof of Theorem 2.13 is given in Section 3.6.

2.5. Uniformly almost periodic functions

Finally, we give an application of Corollary 2.6 (which follows from our main Theorem 2.1) to uniformly almost periodic functions. We show that $H_{\infty}^{\lambda}(G)$ may be identified isometrically and ’coefficient preserving’ with the space of all bounded and holomorphic functions on $[Re>0]$, which are uniformly almost periodic when restricted to any abscissa $[Re=\sigma],\sigma>0$.

Before we state the result, let us recall a few basic definitions and facts from the theory of almost periodic functions – we refer to [2, Chapter III] for more information.

A continuous function $g:\mathbb{R}\to\mathbb{C}$ is said to be uniformly almost periodic (see [2, pp.1-2]) if for every $\varepsilon>0$ there is a number $l>0$ such that for all intervals $I\subset\mathbb{R}$ with $|I|=l$ there is $\tau\in I$ such that

[TABLE]

Equivalently, a continuous function $g\colon\mathbb{R}\to\mathbb{C}$ is uniformly almost periodic if and only if it is the uniform limit of trigonometric polynomials of the form $p(t)=\sum_{n=1}^{N}a_{x_{n}}e^{-itx_{n}}$, where $x_{n}\in\mathbb{R}$ (see [2, $2^{\circ}$ Theorem, p. 29]).

A holomorphic function $F$ defined on an open strip $[\alpha<Re<\beta]$ is said to be uniformly almost periodic (see [2, pp.141-142]) if for every $\varepsilon>0$ there is a number $l>0$ such that for all intervals $I\subset\mathbb{R}$ with $|I|=l$ there is $\tau\in I$ for which

[TABLE]

Obviously, this implies that for every $\alpha<\sigma<\beta$ each of its restrictions $F_{\sigma}=F(\sigma+i\cdot):\mathbb{R}\to\mathbb{C}$ is uniformly almost periodic.

Moreover, in [2, $4^{\circ}$ Theorem, p. 142-143] the following is proved: If a bounded and holomorphic function $F:[Re>0]\to\mathbb{C}$ for some $\sigma_{0}>0$ has a uniformly almost periodic restriction $F_{\sigma_{0}}=F(\sigma_{0}+i\cdot):\mathbb{R}\to\mathbb{C}$, then it is uniformly almost periodic on every possible smaller strip $[\alpha_{1}<Re<\beta_{1}]$ with $0<\alpha_{1}<\beta_{1}<\infty$.

Fixing $x\in\mathbb{R}$ and $\sigma>0$, we call

[TABLE]

the $x$th Bohr coefficient of $F$; it appears that the definition of $a_{x}$ is independent of the choice of $\sigma$ (see [2, p. 147]). Moreover, it can be shown, that only at most countable many Bohr coefficients are non zero, and that $F$ vanishes, whenever its Bohr coefficients vanish (see [2, p. 148 and p. 18]).

Note that finite sums of the form $F(z):=\sum_{n=1}^{N}a_{n}e^{-\lambda_{n}z}$ (all coefficients $\neq 0$) are typical examples of holomorphic, uniformly almost periodic functions, and then the coefficients $a_{k},\,1\leq k\leq N,$ are precisely the (non-zero) Bohr coefficients of $F$.

Definition 2.15**.**

Given a frequency $\lambda$, define $\mathcal{H}_{\infty}^{\lambda}[Re>0]$ to be the space of all bounded, holomorphic functions $F\colon[Re>0]\to\mathbb{C}$, which are uniformly almost periodic on all (or equivalently some) abscissa $[Re=\sigma],\sigma>0$ and for which the Bohr coefficients $a_{x}(F)$ vanish whenever $x\notin\lambda$.

Together with $\|F\|_{\infty}:=\sup_{[Re>0]}|F(z)|$ the space $\mathcal{H}_{\infty}^{\lambda}[Re>0]$ becomes a Banach space, and we have $|a_{x}(F)|\leq\|F\|_{\infty}$ for all $x\in\mathbb{R}$.

Recall that, by [4], if $\lambda$ satisfies $(LC)$, then the space $H_{\infty}^{\lambda}(G)$ equals $\mathcal{D}_{\infty}(\lambda)$, which is the space of all $\lambda$-Dirichlet series $D=\sum a_{n}e^{-\lambda_{n}s}$ converging and defining a bounded function on $[Re>0]$. As proved in [19], such Dirichlet series converge uniformly on ever half-plane $[Re>0],\sigma>0$, which implies that their limit functions belong to $\mathcal{H}_{\infty}^{\lambda}[Re>0]$, i.e. under $(LC)$ the Banach space $H_{\infty}^{\lambda}(G)$ embeds isometrically into $\mathcal{H}_{\infty}^{\lambda}[Re>0]$. Using Theorem 2.1, we can show much more – for any $\lambda$ both spaces may be identified ’coefficient preserving’.

Theorem 2.16**.**

Let $\lambda$ be an arbitrary frequency and $(G,\beta)$ a $\lambda$-Dirichlet group. Then there is an onto isometry

[TABLE]

which preserves Bohr and Fourier coefficients.

Note that Theorem 2.16 again proves that the definition of $H_{\infty}^{\lambda}(G)$ is independent of the chosen $\lambda$-Dirichlet group $(G,\beta)$ (see also [4]).

Remark 2.17**.**

Recall from [19] that the space $\mathcal{D}^{ext}_{\infty}(\lambda)$ denotes the space of all somewhere convergent Dirichlet series, which allow a bounded holomorphic extension to $[Re>0]$. By [4], for any frequency $\lambda$ there is an injective contraction $\Psi_{1}\colon\mathcal{D}^{ext}_{\infty}(\lambda)\hookrightarrow H_{\infty}^{\lambda}(G),\leavevmode\nobreak\ \leavevmode\nobreak\ D\mapsto f$, which preserves the Dirichlet and Fourier coefficients. On the other hand, by [19] the map $\Psi_{2}\colon\mathcal{D}^{ext}_{\infty}(\lambda)\hookrightarrow\mathcal{H}_{\infty}^{\lambda}[Re>0],\leavevmode\nobreak\ \leavevmode\nobreak\ D\mapsto F,$ where $F$ is the limit function of $D$, defines an into isometry. Theorem 2.16 shows that $\Psi_{1}$ is even isometric and that the mapping $\Psi$ from Theorem 2.16 commutes with $\Psi_{1}$ and $\Psi_{2}$, that is we have $\Psi_{1}=\Psi\circ\Psi_{2}.$

Remark 2.18**.**

To every function $F\in\mathcal{H}_{\infty}^{\lambda}[Re>0]$ we may (formally) assign its Dirichlet series $D:=\sum a_{\lambda_{n}}(F)e^{-\lambda_{n}s}$. The question appears, under what conditions on $\lambda$ does $D$ converge on $[Re>0]$. Recall that, if $\lambda$ satisfies Landau’s condition (15), then $D$ even converges uniformly on $[Re>0]$ (see e.g. [19] or [15]).

We finish with the following consequence of Theorem 2.16. By [11] the Hardy space $H_{\infty}(\mathbb{T}^{\infty})$ isometrically equals the Banach space $H_{\infty}(B_{c_{0}})$ of all bounded and holomorphic functions $F$ on the open unit ball $B_{c_{0}}$ of $c_{0}$, identifying Fourier and Taylor coefficients (see [3, Theorem 5.1] for details). Moreover, given $F\in H_{\infty}(B_{c_{0}})$, for every $u>0$ the function

[TABLE]

is continuous and $\widehat{F_{u}}(\alpha)=c_{\alpha}(F)n^{-u}$ whenever $n=\mathfrak{p}^{\alpha}$. Hence, if $f\in H_{\infty}(\mathbb{T}^{\infty})$ and $F\in H_{\infty}(B_{c_{0}})$ are associated, then we have $f*p_{u}=F_{u}$, and so $f*p_{u}\in C(\mathbb{T}^{\infty})$ for every $u>0$. Theorem 2.16 extends this result to arbitrary $\lambda$-Dirichlet groups (compare also with Corollar 2.2 and Theorem 2.13).

Corollary 2.19**.**

Let $(G,\beta)$ be a $\lambda$-Dirichlet group, and $f\in H_{\infty}^{\lambda}(G)$. Then $f*p_{u}\in C(G)$ for all $u>0$, and for all $k>0$

[TABLE]

uniformly on $G$.

The proof of Theorem 2.16 is in Section 3.6.

3. Proofs

The proofs of the results presented in the previous sections are provided according to the following order:

• •

Proof of Theorem 2.10 (Section 3.1)

• •

Proof of Theorem 2.1 (Section 3.2)

• •

Proof of Corollary 2.2, Proposition 2.4, and Corollary 2.11 (Section 3.3)

• •

Proof of Lemma 1.4 and Corollary 2.6 (Section 3.4)

• •

Proof of the Theorems 2.5, 2.7, and 2.9 (Section 3.5)

• •

Proof of the Theorems 2.13, 2.16, and Corollary 2.19 (Section 2.5)

3.1. Proof of Theorem 2.10

Fixing $A>0$, consider the following ’graduated’ variant of $\overline{M}$, that is

[TABLE]

where $f\in L_{1}(G)$. Note that with this definition we have

[TABLE]

Moreover, since for all intervals $I$ the function

[TABLE]

is measurable, by Fubini’s theorem for almost all $\omega\in G$ the function

[TABLE]

is measurable. Recall that the pointwise supremum of a countable family of measurable functions is again measurable. So, if we in the definition from (30) consider all intervals with rational boundary points, then $\overline{M}_{A}(f)$ is measurable and it leads to the same operator. Indeed, if $I$ is an non empty interval, then we are able to choose a sequence of subintervals $I_{n}\subset I$ with rational boundary points, such that \lim_{n\to\infty}\big{|}I\setminus I_{n}\big{|}=0. Then

[TABLE]

and so tending $n\to\infty$ we obtain

[TABLE]

Hence $\overline{M}_{A}(f)$ is measurable for all $A>0$, and by $(\ref{friday})$ we see that $\overline{M}(f)$ is also measurable. Now Theorem 2.10 is a consequence of the following lemma.

Lemma 3.1**.**

Let $\alpha>0$ and $f\in L_{1}(G)$. Then for all $A>0$

[TABLE]

Proof.

Let $A>0$ and fix $\alpha>0$. We define

[TABLE]

Then $\Omega(\alpha)$ is measurable, since $\overline{M}_{A}(f)$ is measurable. Moreover, for $\omega\in G$ we define

[TABLE]

which is Lebesgue-measurable for almost all $\omega\in G$, since $\Omega(\alpha)$ is measurable. Hence by definition for all $t\in[-A,A]$

[TABLE]

and so we obtain for all $A>0$

[TABLE]

where $\eta$ denotes the Lebesgue measure restricted to $[-A,A]$. We now claim that

[TABLE]

Indeed, if this estimate is verified, then we finally obtain

[TABLE]

So let us check (32). For every $t\in\Omega_{\omega}(\alpha)$ (by definition) there is an open interval $I_{t}\in[-2A,2A]$ containing $t$ such that

[TABLE]

By Vitali’s covering theorem (see e.g. [7, Theorem 1.24, p. 36]) there is a sequence $(t_{n})\subset\Omega_{\omega}(\alpha)$ such that

[TABLE]

where the latter union is disjoint. So by $(\ref{standardargument2})$

[TABLE]

Finally, we are ready to give the

Proof of Theorem 2.10.

Take $f\in L_{1}(G)$ and $\alpha>0$, and define for $N\in\mathbb{N}$

[TABLE]

Then

[TABLE]

and, since $\Omega_{N}(\alpha)\subset\Omega_{N+1}(\alpha)$ for all $N$, we by Lemma 3.1 have

[TABLE]

The case $p=\infty$ follows directly from the fact, that $\|f_{\omega}\|_{L_{\infty}(\mathbb{R})}=\|f\|_{L_{\infty}(G)}$ (see [4, Lemma 3.10]). Now Marcinkiewicz’s interpolation theorem (see e.g. [9, Theorem 1.3.2., p. 34]) gives the claim for $1<p<\infty$. ∎

3.2. Proof of Theorem 2.1

The next proposition reduces the proof of Theorem 2.1 to Theorem 2.10.

Proposition 3.2**.**

Let $\lambda$ be a frequency and $(G,\beta)$ a $\lambda$-Dirichlet group. Then for every $k>0$ there is a constant $C(k)>0$ such for all $f\in H_{1}^{\lambda}(G)$ and for almost all $\omega\in G$

[TABLE]

Moreover, for $0<k\leq 1$ the choice $C(k)=Ck^{-1}$ with an absolute constant $C$ is possible .

Before we start to prove this result we show how it gives the

Proof of Theorem 2.1.

Applying the $H_{p}^{\lambda}(G)$-norm to $(\ref{reductioneq})$, Theorem 2.10 shows that $R_{\max}^{\lambda,k}$ defines a bounded operator from $H_{1}^{\lambda}(G)$ to $L_{1,\infty}(G)$, and from $H_{p}^{\lambda}(G)$ to $L_{p}^{\lambda}(G)$ whenever $1<p<\infty$. ∎

We need two ingredients for the proof of Proposition 3.2. The first one is the following integral representation for first Riesz means, where we denote by $\mathcal{F}_{L_{1}(\mathbb{R})}$ the Fourier transform on $L_{1}(\mathbb{R})$.

Lemma 3.3**.**

Let $f\in H_{1}^{\lambda}(G)$ and $0<k\leq 1$. Then we for almost all $\omega\in G$ for all $x>0$ and $u\geq 0$ have

[TABLE]

In order to prove (34), we will see, that in (35) it suffices to have control of the $L_{1}(\mathbb{R})$-norm of the function the Fourier tranform is applied on; our second ingredient for the proof of Proposition 3.2.

Lemma 3.4**.**

Let $v>0$, $0<k\leq 1$ and $u\geq 0$. Then

[TABLE]

Moreover, if $u=0$, then

[TABLE]

We first show, how Lemma 3.3 and Lemma 3.4 imply Proposition 3.2. After that, we give a proof of Lemma 3.3 which uses Lemma 3.4, and eventually we prove Lemma 3.4.

Proof of Proposition 3.2.

In a first step we assume that $0<k\leq 1$. Let

[TABLE]

Then with Lemma 3.3 and Lemma 3.4 we obtain

[TABLE]

Now by [9, Theorem 2.1.10, p.91] we have

[TABLE]

which proves the claim with constant

[TABLE]

If $k>1$, we write $k=l+k^{\prime}$, where $l\in\mathbb{N}$ and $k^{\prime}\in]0,1]$, and use for every $x>0$ the following identity from [10, Lemma 6, p. 27]:

[TABLE]

where by substitution (see [10, p. 27])

[TABLE]

Together this leads to

[TABLE]

which, applying the first step with $0<k^{\prime}\leq 1$, finishes the proof. ∎

Proof of Lemma 3.3.

Fix $u\geq 0$ and let first $f=\sum_{n=1}^{N}a_{n}h_{\lambda_{n}}$. Then $f_{\omega}(t)=\sum_{n=1}^{N}a_{n}h_{\lambda_{n}}(\omega)e^{-it\lambda_{n}}$ for all $\omega\in G$ and, since for all $\alpha>0$ and $k>0$

[TABLE]

(see e.g. [10, Lemma 10, p. 50]), we for all $c>u$, $x>0$ obtain (with $\alpha=c-u$)

[TABLE]

The choice $c=\frac{1}{x}+u$ leads to

[TABLE]

and so the claim holds for polynomials in $H_{1}^{\lambda}(G)$. To proof the general case, observe that for all $u\geq 0$ and $v>0$ the operator

[TABLE]

is bounded. Indeed, by Lemma 3.4 (and Fubini’s theorem) we have

[TABLE]

Additionally, this shows, that $\frac{f_{\omega}*P_{u+v}}{(v+i\cdot)^{1+k}}\in L_{1}(\mathbb{R})$ for almost all $\omega\in G$, and so we in particular obtain (with Fubini’s theorem)

[TABLE]

Now let $(Q^{n})\subset H_{1}^{\lambda}(G)$ be a sequence of polynomials converging to $f$ in $H_{1}^{\lambda}(G)$ (see [4]). Then, by continuity of $\mathcal{A}$ and $\mathcal{F}_{L_{1}(\mathbb{R})}$, we for almost all $\omega\in G$ obtain

[TABLE]

for some subsequence $(Q^{n_{k}})$, with uniformly convergence on $\mathbb{R}$. So together with (41) and (43)

[TABLE]

which gives the claim by (43). ∎

Proof of Lemma 3.4.

By substitution we have

[TABLE]

We interpret the right hand side of $(\ref{calcusubst})$ as the $L_{1+k}$-norm with respect to the measure $d\mu=P_{u+v}d\lambda$, where $d\lambda$ denotes the Lebesgue measure on $\mathbb{R}$. Since $(\mathbb{R},d\mu)$ is a finite measure space, we for all $0\leq k\leq 1$ have

[TABLE]

Hence, it suffices to determine $(\ref{calcusubst})$ for $k=1$. In this case, a straight calculation gives

[TABLE]

So, if $u=0$, then we have

[TABLE]

and, if $u\neq 0$, then we estimate

[TABLE]

Now by taking the $\frac{1+k}{2}$th power the claims follow. ∎

3.3. Proof of Corollary 2.2, Proposition 2.4, and Corollary 2.11

It is quite standard to deduce almost everywhere convergence from appropriate maximal inequalities. Nevertheless we for the sake of completeness add a few details. We use the following standard consequence of Egoroff’s theorem (see e.g. [18, Theorem 4.4, p. 33]).

Remark 3.5**.**

Let $f_{n}:\Omega\to\mathbb{C}$ be measurable functions on a finite measure space $(\Omega,\mu)$. Then $(f_{n})$ converges to [math] almost everywhere if and only if for every $\varepsilon>0$ we have

[TABLE]

The following device adapts some well-known arguments to our special situation.

Lemma 3.6**.**

Let $X$ be a subspace of $L_{1,\infty}(\mu)$, and \big{(}T_{x,y}\colon X\to L_{1}(\mu)\big{)}_{x,y>0} and \big{(}S_{y}\colon Y\to L_{1,\infty}(\mu)\big{)}_{y>0} two families of linear operators such that the sublinear maps

[TABLE]

are bounded. Moreover, let $Y$ be a dense subset of $X$ such that for all $g\in Y$

[TABLE]

Then this equation even holds for all $f\in X$.

Proof.

Let $f\in X$ and $\varepsilon>0$. According to Remark 3.5 we show that

[TABLE]

Denote the set which appears on the left side by $\Omega$, and use the assumption on $Y$ to conclude for all $\omega\in\Omega$ and $g\in Y$

[TABLE]

Hence the boundedness of $T$ and $S_{y}$ (and the quasi triangle inequality in $L_{1,\infty}(\mu)$) gives some $C>0$ such that for all $g\in Y$

[TABLE]

Finally, the density of $Y$ in $X$ gives the conclusion. ∎

We will also need the following ’shifted’ version of Theorem 2.1.

Proposition 3.7**.**

Let $f\in H_{1}^{\lambda}(G)$ and $k>0$. Then

[TABLE]

Proof.

By Proposition 3.2 and Theorem 2.1, it suffices to show, that

[TABLE]

Indeed, for all intervals $I\subset\mathbb{R}$ and $u>0$ we have (using Fubini’s theorem)

[TABLE]

and so, since the ’restriction’ of $f*p_{u}$ to $\mathbb{R}$ is given by the function $f_{\omega}*P_{u}$, we for almost all $\omega\in G$ have

[TABLE]

Applying the $L_{1,\infty}$-norm gives the inequality we claimed. ∎

Proof of Corollary 2.2.

Let us first proof (19). Take $X=H_{1}^{\lambda}(G)$, $T_{x,y}(f):=R_{x}^{\lambda,k}(f)$ and $S_{y}$ the identity map. Then $\lim_{x\to\infty}R_{x}^{\lambda,k}(P)=P$ pointwise for all polynomials from $H_{1}^{\lambda}(G)$, and so the claim follows from Lemma 3.6 and Theorem 2.1. The proof of (20) is similar and needs Proposition 3.7. ∎

Proof of Proposition 2.4.

We first show, that there is a null set $N\subset G$ such that for all $\omega\notin N$ the integral

[TABLE]

is finite for all $u>0$. Indeed, recall from Section 1.1 that $f_{\omega}$ is locally Lebesque-integrable for almost all $\omega$, and so by [9, Theorem 2.1.10, p. 91] we for all $u>0$ and almost all $\omega$ have

[TABLE]

Since $\overline{M}(f)\in L_{1,\infty}(G)$ by Theorem 2.10, we obtain that $\overline{M}(f)(\omega)<\infty$ almost everywhere and that the operator $T$ is defined. Moreover, $|T(f)(\omega)|\leq\overline{M}(f)(\omega)$ for almost all $\omega$, and again Theorem 2.10 implies that $T$ is bounded from $L_{1}(G)$ to $L_{1,\infty}(G)$ and $L_{p}(G)$ to $L_{p}(G)$, whenever $1<p\leq\infty$. The ’in particular’ is then a consequence of Lemma 3.6 with the choice $T_{x}(f)=f*p_{x}$, $S_{y}\equiv id$, and $Y$ the set of all polynomials. ∎

Proof of Corollary 2.11.

Equations (27) and (28) are checked straight forward on polynomials. Then both claims follow from Lemma 3.6 and Theorem 2.10 by choosing $X=L_{1}(G)$ and $T_{x,y}(f)=\frac{1}{2x}\int_{-x}^{x}f_{\omega}(t)dt$ (resp. $\widetilde{T}_{x,y}(f):=T_{x^{-1},y}(f)$), since clearly $T_{x,y}(f)(\omega)\leq\overline{M}(f)(\omega)$ for all $x,y$. ∎

3.4. Proof of Lemma 1.4 and Corollary 2.6

We start with the proof of Lemma 1.4, which shows how Riesz-summability of the Fourier series of a function $f\in H_{p}^{\lambda}(G)$ transfers to summability of the vertical limit of $D^{\omega}$, where $D:=\mathcal{B}(f)$, and vice versa.

Proof of Lemma 1.4.

It suffices to check that $(1)$ and $(2)$ are equivalent. Given a measurable set $A\subset G$ for almost every $\omega\in G$ the set

[TABLE]

is Lebesgue-measurable and by Fubini’s theorem we have

[TABLE]

where $\widetilde{\lambda}=(1+t^{2})^{-1}dt.$ Hence if, given $\varepsilon>0$, we define

[TABLE]

then

[TABLE]

and so

[TABLE]

By Remark 3.5, assuming $(1)$, the left hand side of (46) vanishes, and so for almost all $\omega\in G$ we for almost all $t\in\mathbb{R}$ have

[TABLE]

and so equivalently

[TABLE]

Vice versa, assuming (2), the right hand side of (46) vanishes, and so (1) follows from Remark 3.5. ∎

Proof of Corollary 2.6.

Translate Corollary 2.2 with the help of Lemma 1.4 into Dirichlet series. ∎

3.5. Proof of the Theorems 2.5, 2.7

and 2.9

For the proof of Theorem 2.5 we need the following

Lemma 3.8**.**

Let $0<k\leq 1$ and $u>0$. Then for every $\varepsilon>0$ there is a constant $C=(k,u,\varepsilon)$ such that for all $x>0$ and complex sequences $(a_{n})$ we have

[TABLE]

The proof of Lemma 3.8 follows from a careful analysis of Theorem 24 from Hardy and M. Riesz [10, §VI.3, p. 42 ]. Among others, we use the following identity from [10, §IV.2, p. 21]:

[TABLE]

Moreover, in the case $0<k<1$, we need the following two integrals

[TABLE]

and

[TABLE]

the first follows by simple substitution using the beta function and the second one is obvious.

Proof of Lemma 3.8.

Let us write for simplicity

[TABLE]

Then, defining $h(t):=(e^{-(u+\varepsilon)t}-e^{-(u+\varepsilon)x})(x-t)^{k}$ for $0<t<x$, we obtain

[TABLE]

where the first equality follows from Abel summation (see [10, p. 40]), and the third by (47) and partial integration, since

[TABLE]

Now let first $0<k<1$. Then by [10, Lemma 6, p.27] we have

[TABLE]

Then Fubini’s theorem implies

[TABLE]

where

[TABLE]

Using $(e^{-(u+\varepsilon)t}-e^{-(u+\varepsilon)x})\leq e^{-(u+\varepsilon)t}(u+\varepsilon)(x-t)$ for the third summand we estimate

[TABLE]

Then we deduce from (48) and (49) that for all $y>0$

[TABLE]

Hence finally

[TABLE]

Note, that the case $k=1$ follows the same lines with the difference, that we do not use (51) and estimate $|B|$ directly. ∎

Proof of Theorem 2.5.

Fix $k,u>0$, and assume first that $0<k\leq 1$. Then by Lemma 3.8 it suffices to prove that for all $f\in H_{p}^{\lambda}(G)$

[TABLE]

Let first $f=\sum_{n=1}^{N}a_{n}h_{\lambda_{n}}$ be a polynomial. Then applying [19, Lemma 3.6] (with $\varepsilon=u$ and $D^{\omega}=\sum_{n=1}^{N}a_{n}h_{\lambda_{n}}(\omega)h_{\lambda_{n}}$, or using again (40) straight away) we obtain for all $x>0$ and $\omega\in G$

[TABLE]

Like in the proof of Lemma 3.3 the continuity of $\mathcal{A}$ from (42) as well as the continuity of the Fourier transform $\mathcal{F}_{L_{1}(\mathbb{R})}$ imply that (54) holds for every $f\in H_{1}^{\lambda}(G)$, all $x>0$, and almost all $\omega\in G$. Hence for such $f,x$ and $\omega$

[TABLE]

where we used Lemma 3.4 for the last inequality. Now integration over $G$ and the Minkowski inequality give for all $f\in H_{p}^{\lambda}(G)$

[TABLE]

which under the restriction that $0<k\leq 1$ is what we aimed at in (53). If $k>1$, then we write $k=l+k^{\prime}$, where $l\in\mathbb{N}$ and $0<k^{\prime}\leq 1$. Replacing $f$ by $f*p_{u}$ in (39) we conclude that

[TABLE]

which proves the claim for all $k>0$. ∎

Proof of Theorem 2.7.

Combine (28) from Corollary 2.11 with Theorem 2.5. ∎

Using the next lemma, Theorem 2.9 follows from Theorem 2.5.

Lemma 3.9**.**

Let $0<k\leq 1$ and $u>0$. Then for every $\varepsilon>0$ there is a constant $C=C(k,u,\varepsilon)$ such that for all $x>0$ and complex sequences $(a_{n})$ we have

[TABLE]

We follow a similar strategy as in the previous proof of Lemma 3.8, use the following identity from [10, §IV.2, p. 21]

[TABLE]

and also some ideas from [10, Proof of Theorem 20, p. 33].

Proof of Lemma 3.9.

Let $u,\varepsilon>0$. By Lemma 3.8 it suffices to prove

[TABLE]

Moreover, let us for simplicity write

[TABLE]

We use the following identity (see again the beginning of the proof of Lemma 3.8)

[TABLE]

where $h(l):=(l^{-\varepsilon}-e^{-\varepsilon x})(e^{x}-l)^{k}$. Let us first deal with the summand A. By substitution with $t=\log(l)$ we obtain

[TABLE]

Since the positive function $G(t):=\left(\frac{e^{x}-e^{t}}{x-t}\right)^{k-1}e^{t}$, where $0<t<x$, is increasing with $\lim_{r\to x}G(r)=e^{xk}$, by the second mean value theorem (applied separately to the real and imaginary part) there are $0<\xi_{1},\xi_{2}<x$ such that

[TABLE]

and so by [10, Lemma 7, p. 28]

[TABLE]

Now we consider the second summand $B$, and define $g(t):=h^{\prime}(e^{t})e^{t}$, where $0<t<x$. Then the substitution $t=\log(l)$ and partial integration (use again (50)) give

[TABLE]

Let now $0<k<1$. Then using (51) and Fubini’s theorem we finally end up with

[TABLE]

where

[TABLE]

Estimating straight forward there is a constant $C_{2}=C_{2}(k,\varepsilon)$ such that

[TABLE]

Hence, following the estimates from the end of the proof of Lemma 3.8 (compare the bound for $|g^{\prime}(t)|$ with the bound for $|h^{\prime\prime}(t)|$ from (52)) we conclude that

[TABLE]

Finally (56) follows, since

[TABLE]

Note that the case $k=1$ again follows the same lines without using (51). ∎

Proof of Theorem 2.9.

Lemma 3.9 and (53) assure that for $0<k\leq 1$ we have

[TABLE]

So let $k>1$ and write $k=l+k^{\prime}$, where $l\in\mathbb{N}$ and $0<k^{\prime}\leq 1$. It suffices to show that

[TABLE]

Indeed, by definition and (39) we have

[TABLE]

3.6. Proof of the Theorems 2.13 and 2.16, and Corollary 2.19

The following observation is an important tool of both proofs.

Lemma 3.10**.**

Let $\lambda$ be a frequency, $k>0$ and $(G,\beta)$ a $\lambda$-Dirichlet group. Then there is a constant $C>0$ such that for all $x>0$ there is a measure $\mu_{x}\in M(G)$ with $\|\mu_{x}\|\leq C$ and such that for all $n\in\mathbb{N}$

[TABLE]

Proof of Lemma 3.10.

The case $p=\infty$ of Theorem 2.1 implies that there is a constant $C>0$ such that for all $x>0$ and all $f\in H_{\infty}^{\lambda}(G)$

[TABLE]

Denote the subspace of all continuous functions in $H_{\infty}^{\lambda}(G)$ by $C^{\lambda}(G)$, and fix some $x>0$. Then the bounded functional

[TABLE]

has norm $\leq C$, and satisfies T_{x}(h_{\lambda_{n}})=\big{(}1-\frac{\lambda_{n}}{x}\big{)}^{k} for $\lambda_{n}<x$ and $T_{x}(h_{\lambda_{n}})=0$ for $\lambda_{n}\geq x$. By the Hahn-Banach theorem there is $\widetilde{T_{x}}\in(C(G))^{\prime}$ extending $T_{x}$ with equal norm, and then also the linear operator

[TABLE]

has norm $\leq C$, and satisfies R_{x}(\overline{h_{\lambda_{n}}})=\big{(}1-\frac{\lambda_{n}}{x}\big{)}^{k} for $\lambda_{n}<x$ and $R_{x}(\overline{h_{\lambda_{n}}})=0$ for $\lambda_{n}\geq x$. Hence the Riesz representation theorem assures the existence of a measure $\mu_{x}$ with norm $\leq C$ which, since $\widehat{\mu_{x}}(h_{\lambda_{n}})=R_{x}(\overline{h_{\lambda_{n}}})$ for all $n$, has the desired Fourier coefficients. ∎

Proof of Theorem 2.13.

Note that for any polynomial $P\in H_{1}^{\lambda}(G)$ we have

[TABLE]

where $\mu_{x}$ is the measure from Lemma 3.10. Now, given $f\in H_{1}^{\lambda}(G)$ and $\varepsilon>0$, choose by density a polynomial $P$ such that $\|f-P\|_{1}\leq\varepsilon$. Then for large $x$ (and the constant $C$ from Lemma 3.10)

[TABLE]

Observe, that the counterexamples of Remark 2.14 show that the variant of Lemma 3.10 for second Riesz means does not hold in the sense that there are no measures $\mu_{x}\in M(G)$, $x>0$, $\|\mu_{x}\|\leq C$ for some $C>0$, such that

[TABLE]

In the proof of Theorem 2.16 we take advantage of Lemma 3.10 and combine it with an estimate for the abscissa of uniform summability by Riesz means. Given $D=\sum a_{n}e^{-\lambda_{n}s}$ and $k\geq 0$, we define $\sigma_{u}^{\lambda,k}(D)$ to be the infimum of all $\sigma\in\mathbb{R}$ such that $D$ is uniformly $(\lambda,k)$-summable on $[Re>\sigma]$, i.e. the limit

[TABLE]

exists uniformly on $[Re>\sigma]$. We are going to make use of the following Bohr-Cahen type formula proved in [19]:

[TABLE]

with equality whenever $\sigma_{u}^{\lambda,k}(D)\geq 0$.

Proof of Theorem 2.16.

Let us start defining a contractive coefficient preserving mapping

[TABLE]

Take $F\in\mathcal{H}_{\infty}^{\lambda}[Re>0]$. Then (as described in Section 2.5), given $\sigma>0$, the uniformly almost periodic function $F_{\sigma}=F(\sigma+i\cdot)$ is a uniform limits of polynomials of the form $P^{N}_{\sigma}(t)=\sum_{n}b_{n}^{N}e^{-\lambda_{n}it}$. Hence by density of $\beta:\mathbb{R}\to G$, the polynomials $p_{\sigma}^{N}:=\sum_{n}b_{n}^{N}h_{\lambda_{n}}$ form a Cauchy sequence in $H_{\infty}^{\lambda}(G)$ with limit, say, $f_{\sigma}$ with $\|f_{\sigma}\|_{\infty}=\|F_{\sigma}\|_{\infty}\leq\|F\|_{\infty}$. Then by a standard weak compactness argument there is $\Psi(h):=f\in H_{\infty}^{\lambda}(G)\subset L_{\infty}(G)$ with $\|f\|_{\infty}\leq\|F\|_{\infty}$, which is the weak star limit of some subsequence of $(f_{\frac{1}{n}})_{n}$ (use that the unit ball of $L_{\infty}(G)$, being the dual of $L_{1}(G)$, endowed with its weak star topology is compact and metrizable). Then a simple argument shows that $a_{\lambda_{n}}(F)=\widehat{f}(h_{\lambda_{n}})$ for all $n$, i.e. $\Psi$ is indeed an coefficient preserving contraction.

In order to show that $\Psi$ is in fact an isometry onto, take $f\in H_{\infty}^{\lambda}(G)$. Using the measures $\mu_{x}$ from Lemma 3.10 and the fact that $\beta$ has dense range, we for all $x>0$ have

[TABLE]

Hence (58) applied to $D:=\sum\widehat{f}(h_{\lambda_{n}})e^{-\lambda_{n}s}$ shows that $\sigma_{u}^{\lambda,1}(D)\leq 0$, and this in particular proves that

[TABLE]

defines a holomorphic function on $[Re>0]$ which converges uniformly on every smaller half-space $[Re>\sigma],\sigma>0$. As explained in Section 2.5 we may deduce that all functions $F_{\sigma}=F(\sigma+i\cdot),\,\sigma>0$ are uniformly almost periodic with Bohr coefficients $a_{\lambda_{n}}(F)=\widehat{f}(h_{\lambda_{n}})$ for all $n$ and zero else. It remains to show, that $F$ is bounded. By equation (25) from Corollary 2.6 there is some $\omega\in G$, such that for all $\sigma>0$ and almost all $t\in\mathbb{R}$ we have

[TABLE]

note that here both sides form continuous functions, and hence the equality in fact holds for every $t\in\mathbb{R}$. On the other hand we deduce from the rotation invariance of the Haar measure that

[TABLE]

and therefore another application of (59) and (58) shows that the vertical limits $D^{\omega}=\sum\widehat{f}(h_{\lambda_{n}})h_{\lambda_{n}}(\omega)e^{-\lambda_{n}s}$ are uniformly summable by first $(\lambda,1)$-Riesz means on all half-planes $[Re>\sigma]$ with $\sigma>0$. All together this implies

[TABLE]

and so $\Psi$ is indeed an isometry onto. ∎

Proof of Corollary 2.19.

Let $f\in H_{\infty}^{\lambda}(G)$ and $F:=\Psi^{-1}(f)$, where $\Psi$ is the mapping from Theorem 2.16. Then for every $u>0$ the restriction $F_{u}=F(u+i\cdot)$ is uniformly almost periodic on $\mathbb{R}$. So there is $g_{u}\in C(G)\cap H_{\infty}^{\lambda}(G)$ such that $F_{u}=g_{u}\circ\beta$, and $\widehat{g_{u}}(h_{\lambda_{n}})=a_{\lambda_{n}}(F)e^{-u\lambda_{n}}=\widehat{f}(h_{\lambda_{n}})e^{-u\lambda_{n}}$ for all $n$. Hence $g_{u}=f*p_{u}$ (compare Fourier coefficients). The second statement follows by approximation with polynomials of the form $\sum_{n=1}^{N}b_{n}h_{\lambda_{n}}$, which are dense in $C(G)\cap H_{\infty}^{\lambda}(G)$ (see [17, §8.7.3]). ∎