Volterra operators and Hankel forms on Bergman spaces of Dirichlet series

Abstract

For a Dirichlet series g, we study the Volterra operator $T_g f(s)=-\int ^{+\infty }_{s} f(w)g'(w)dw,$ acting on a class of weighted Hilbert spaces ${{\mathcal {H}}^{2}_{w}}$ of Dirichlet series. We obtain sufficient / necessary conditions for $T_g$ to be bounded (resp. compact), involving BMO and Bloch type spaces on some half-plane. We also investigate the membership of $T_g$ in Schatten classes. Moreover, we show that if $T_g$ is bounded, then g is in ${{\mathcal {H}}}^p_w$, the $L^p$-version of ${{\mathcal {H}}^{2}_{w}}$, for every $0<p<\infty$. We also relate the boundedness of $T_g$ to the boundedness of a multiplicative Hankel form of symbol g, and the membership of g in the dual of ${{\mathcal {H}}}^1_w$.

1 Introduction

Dirichlet series are functions of the form

$$\begin{aligned} f(s)= \sum ^{+\infty }_{n=1}a_n n^{-s},\ \text { with }s\in \mathbb {C}. \end{aligned}$$

(1.1)

For a real number $\theta$, $\mathbb {C}_{\theta }$ stands for the half-plane $\left\{ s,\ \mathfrak {R}s>\theta \right\}$, and $\mathbb {D}$ for the unit disk. ${{\mathcal {D}}}$ denotes the class of functions f of the form (1.1) in some half-plane $\mathbb {C}_{\theta }$, and ${{\mathcal {P}}}$ is the space of Dirichlet polynomials.

The increasing sequence of prime numbers will be denoted by $(p_j)_{j\ge 1}$, and the set of all primes by $\mathbb {P}$. Given a positive integer n, $n=p^{\kappa }$ will stand for the prime number factorization $n=p^{\kappa _1}_{1}p^{\kappa _2}_{2}\cdots p^{\kappa _d}_{d},$ which associates uniquely to n the finite multi-index $\kappa (n)=(\kappa _1,\kappa _2,\ldots ,\kappa _d)$. The number of prime factors in n is denoted by $\Omega (n)$ (counting multiplicities), and by $\omega (n)$ (without multiplicities).

The space of eventually zero complex sequences $c_{00}$ consists in all sequences which have only finitely many non zero elements. We set $\mathbb {D}^{\infty }_{\text {fin}}=\mathbb {D}^{\infty }\cap c_{00}$ and $\mathbb {N}^{\infty }_{\text {0,fin}}=\mathbb {N}_0^{\infty }\cap c_{00}$, where $\mathbb {N}_0=\mathbb {N}\cup \left\{ 0\right\}$ is the set of non-negative integers.

Let $F:\mathbb {D}^{\infty }_{\text {fin}}\rightarrow \mathbb {C}$ be analytic, i.e. analytic at every point $z\in \mathbb {D}^{\infty }_{\text {fin}}$ separately with respect to each variable. Then F can be written as a convergent Taylor series

$$\begin{aligned} F(z)=\sum _{\alpha \in \mathbb {N}^{\infty }_{\text {0,fin}}}c_{\alpha } z^{\alpha },\ z\in \mathbb {D}^{\infty }_{\text {fin}}. \end{aligned}$$

The truncation $A_m F$ of F onto the first m variables is defined by

$$\begin{aligned} A_m F(z)=F(z_1,\ldots ,z_m,0,0,\ldots ). \end{aligned}$$

For $z,\chi$ in $\mathbb {D}^{\infty }$, we set $z.\chi :=(z_1\chi _1,z_2\chi _2,\ldots )$, and ${{\mathfrak {p}}}^{\mathbf{x}}:=(p_1^x, p_2^x,\ldots )$ for a real number x, .

The Bohr lift [11] of the Dirichlet series $f(s)= \sum ^{+\infty }_{n=1}a_n n^{-s}$ is the power series

$$\begin{aligned} {\mathcal {B}}f(\chi )=\sum ^{+\infty }_{n=1}a_n \chi ^{\kappa (n)}=\sum _{\alpha \in \mathbb {N}^{\infty }_{\text {0,fin}}}{\tilde{a}}_{\alpha } \chi ^{\alpha } ,\ \text { where }{\tilde{a}}_{\alpha }=a_{p^{\alpha }}, \chi \in \mathbb {D}^{\infty }_{\text {fin}}, \end{aligned}$$

with the multiindex notation $\chi ^{\alpha }=\chi _1^{\alpha _1}\chi _2^{\alpha _2}\cdots .$

Given a sequence of positive numbers $w=(w_n)_n=\left( w(n)\right) _n$, one considers the Hilbert space (see [21, 23])

$$\begin{aligned} {{\mathcal {H}}}^2_w:=\left\{ \sum ^{+\infty }_{n=1}a_n n^{-s}:\ \sum ^{+\infty }_{n=1}\frac{\left| a_n\right| ^2}{w_{n}}<+\infty \right\} . \end{aligned}$$

The choice $w_n=1$ corresponds to the space ${{\mathcal {H}}}^2$, introduced in [19].

The weights considered in this article satisfy $w_n=O(n^{\epsilon })$ for every $\epsilon >0$; from the Cauchy-Schwarz inequality, Dirichlet series in ${{\mathcal {H}}^{2}_{w}}$ absolutely converge in $\mathbb {C}_{1/2}$.

We are interested in the Volterra operator $T_g$ of symbol $g(s)=\sum ^{+\infty }_{n=1} b_n n^{-s}$, defined by

$$\begin{aligned} T_g f(s):=-\int ^{+\infty }_{s} f(w)g'(w)dw,\ \mathfrak {R}s>\frac{1}{2}. \end{aligned}$$

(1.2)

On the unit disk $\mathbb {D},$ the Volterra operator, whose symbol is an analytic function g, is given by

$$\begin{aligned} J_gf(z):=\int ^{z}_{0}f(u)g'(u)du,\ z\in \mathbb {D}. \end{aligned}$$

(1.3)

Pommerenke [26] showed that $J_g$ (1.3) is bounded on the Hardy space $H^2(\mathbb {D})$ if and only if g is in $BMOA(\mathbb {D})$. Let $\sigma$ be the Haar measure on the unit circle $\mathbb {T}$. Fefferman’s duality Theorem states that $BMOA(\mathbb {D})$ is the dual space of $H^1(\mathbb {D})$. Thus the boundedness of $J_g$ is equivalent to the boundedness of the Hankel form

$$\begin{aligned} H_g(f,h):=\int _{\mathbb {T}}f(u)h(u)\overline{g(u)}d\sigma (u),\ f,h\in H^2(\mathbb {D}). \end{aligned}$$

(1.4)

Let V be the Lebesgue measure on $\mathbb {C}$, normalized such that $V(\mathbb {D})=1$.

Many authors, in particular [2], have studied Volterra operators on Bergman spaces of $\mathbb {D}$. The classical Bergman space $A^2_{\gamma }(\mathbb {D})$, $\gamma >0$, is associated to the measure $d\tilde{m}_{\gamma }(z):=\gamma \left( 1-\left| z\right| ^2\right) ^{\gamma -1}dV(z)$. $J_g$ is bounded on $A^2_{\gamma }(\mathbb {D})$ if and only if g is in the Bloch space, which is the dual of $A^1_{\gamma }(\mathbb {D})$.

The Bergman space of the finite polydisk $A^2_{\gamma }(\mathbb {D}^d)$, $d\ge 1$, corresponds to the measure

$$\begin{aligned} d\widetilde{\nu }_{\gamma }(z):=d\tilde{m}_{\gamma }(z_1)\times \cdots \times d\tilde{m}_{\gamma }(z_d). \end{aligned}$$

The boundedness of the Hankel form

$$\begin{aligned} H_g(f,h):=\int _{\mathbb {D}^d}f(z)h(z)\overline{g(z)}d\widetilde{\nu }_{\gamma }(z),\ f,h\in A^2_{\gamma }(\mathbb {D}^d), \end{aligned}$$

(1.5)

is equivalent to the membership of g to the Bloch space (see [17]), defined by

$$\begin{aligned} \text {Bloch}(\mathbb {D}^d):=\left\{ f:\mathbb {D}^d\rightarrow \mathbb {C}\text { holomorphic : }\max _{\kappa \in {{\mathcal {I}}}_d}\sup _{z\in \mathbb {D}^d}\left| \partial ^{\kappa }f\left( \kappa .z\right) \right| \left( 1-\left| z\right| \right) ^{\kappa }<+\infty \right\} , \end{aligned}$$

where ${{\mathcal {I}}}_d$ denotes the set of multi-indices $\kappa =\left( \kappa _1,\ldots , \kappa _d\right)$, with entries in $\left\{ 0,1\right\}$, and

$$\begin{aligned} z=\left( z_1,\ldots ,z_d\right) ,\ \partial ^{\kappa }=\partial ^{\kappa _1}_{z_1}\cdots \partial ^{\kappa _d}_{z_d},\ \left( 1-\left| z\right| \right) ^{\kappa }= \left( 1-\left| z_1\right| \right) ^{\kappa _1} \cdots \left( 1-\left| z_d\right| \right) ^{\kappa _d}. \end{aligned}$$

Recall that for $0<p<\infty$, the Hardy space of Dirichlet series ${{\mathcal {H}}}^p$ is the space of Dirichlet series $f \in {{\mathcal {D}}}$ such that ${{\mathcal {B}}}f$ is in $H^p(\mathbb {D}^{\infty })$, endowed with the norm

$$\begin{aligned} \left\| f\right\| _{{{\mathcal {H}}}^p}:=\left\| {{\mathcal {B}}}f\right\| _{H^p(\mathbb {D}^{\infty })}=\left( \int _{\mathbb {T}^{\infty }}\left| {{\mathcal {B}}}f(z)\right| ^p d\sigma _{\infty }(z)\right) ^{1/p}, \end{aligned}$$

$\sigma _{\infty }$ being the Haar measure of the infinite polytorus $\mathbb {T}^{\infty }$.

The norm in the space ${{\mathcal {H}}}^{\infty }:=H^{\infty }(\mathbb {C}_0)\cap {{\mathcal {D}}}$ is

$$\begin{aligned} \left\| f\right\| _{{{\mathcal {H}}}^{\infty }}=\sup _{s\in \mathbb {C}_0}\left| f(s)\right| . \end{aligned}$$

Let $H^{\infty }(\mathbb {D}^{\infty })$ be the space of series F which are finitely bounded, i.e.

$$\begin{aligned} \left\| F\right\| _{H^{\infty }(\mathbb {D}^{\infty })}=\sup _{m\in \mathbb {N}_0,z\in \mathbb {D}^{\infty } }\left| A_m F(z)\right| <\infty . \end{aligned}$$

Via the Bohr isomorphism, we have [16, 19]

$$\begin{aligned} \left\| f\right\| _{{{\mathcal {H}}}^{\infty }}=\left\| {{\mathcal {B}}}f\right\| _{H^{\infty }(\mathbb {D}^{\infty })}. \end{aligned}$$

(1.6)

Several abscissae are related to a function g in ${{\mathcal {D}}}$, of the form $g(s)=\sum ^{+\infty }_{n=1}b_n n^{-s}$:

$$\begin{aligned}&\text {the abscissa of convergence } \sigma _c=\inf \left\{ \sigma \in \mathbb {R}\ :\ \sum \nolimits ^{+\infty }_{n=1}b_n n^{-\sigma }\text { converges }\right\} ;\\&\text {the abscissa of absolute convergence } \sigma _a=\inf \left\{ \sigma \in \mathbb {R}\ :\ \sum \nolimits ^{+\infty }_{n=1}\left| b_n \right| n^{-\sigma }\text { converges }\right\} ;\\&\text {the abscissa of uniform convergence }\\&\sigma _u=\inf \left\{ \theta \in \mathbb {R}\ :\ \sum \nolimits ^{+\infty }_{n=1}b_n n^{-s}\text { converges uniformly in }\mathbb {C}_{\theta }\right\} . \end{aligned}$$

The abscissa of regularity and boundedness, denoted by $\sigma _b$, is the infimum of those $\theta$ such that g(s) has a bounded analytic continuation, to the half-plane $\mathfrak {R}(s)>\theta +\epsilon$, for every $\epsilon >0.$

We have $-\infty \le \sigma _c\le \sigma _u\le \sigma _a\le +\infty$, and, if any of the abscissae is finite $\sigma _a-\sigma _c\le 1$. Moreover, it is known that $\sigma _b=\sigma _u$ [11] , and $\sigma _a-\sigma _u\le \frac{1}{2}$.

Volterra operators (1.2) on the spaces ${{\mathcal {H}}}^p$ have been investigated in [13]. Our aim is to study similar questions for the spaces ${{\mathcal {H}}^{2}_{w}}$, associated to specific weights w in the class ${{\mathcal {W}}}$ defined below.

Definition 1

Let $\beta >0$. A sequence w belongs to ${{\mathcal {W}}}$ if it has one of the following forms:

• (1) $w_n=[d(n)]^{\beta }$, where d(n) is the number of divisors of the integer n. Then ${{\mathcal {H}}^{2}_{w}}:={{\mathcal {B}}^{2}_{\beta }}$.

• (2) $w_n=d_{\beta +1}(n)$, where $d_{\gamma }(n)$ are the Dirichlet coefficients of the power of the Riemann zeta function, namely $\zeta ^{\gamma }(s)=\sum ^{+\infty }_{n=1}d_{\gamma }(n)n^{-s} .$ Then ${{\mathcal {H}}^{2}_{w}}:={{\mathcal {A}}^{2}_{\beta }}$.

As in the case of ${{\mathcal {H}}}^2$ [13], we obtain sufficient/necessary conditions for $T_g$ to be bounded on the Hilbert spaces ${{\mathcal {H}}^{2}_{w}}$. However, due to the lack of information of the behavior of the symbols in the strip $0< \mathfrak {R}s <1/2$, it seems difficult to get an “ if and only if” condition. In the Hardy space setting, it is shown that $T_g$ is bounded on ${{\mathcal {H}}}^2$ provided that g in $BMOA(\mathbb {C}_{0})$. Since the spaces ${{\mathcal {A}}^{2}_{\beta }}$ and ${{\mathcal {B}}^{2}_{\beta }}$ (see Sect. 2) locally behave like Bergman spaces of the half plane $\mathbb {C}_0$, we would expect that the membership of g in $\text {Bloch}(\mathbb {C}_{0})$ (resp. $\text {Bloch}_0(\mathbb {C}_{0})$) would imply the boundedness (resp. compactness) of $T_g$ on ${{\mathcal {H}}}^{2}_{w}.$ We obtain such a sufficient condition when ${{\mathcal {B}}}g$ depends on a finite number of variables $z_1,\ldots ,z_d$. However, our method specfically uses that d is finite, and we do not know whether the same result holds if ${{\mathcal {B}}}g$ is a function of infinitely many variables.

Le ${{\mathfrak {N}}}_d$ be the set of positive integers which are multiples of the primes $p_1,\ldots ,p_d$,

$$\begin{aligned} {{\mathcal {D}}}_d:=\left\{ f\in {{\mathcal {D}}}\ :\ f(s)=\sum _{n\in {{\mathfrak {N}}}_d}a_n n^{-s}\right\} ,\ \text { and }{{\mathcal {H}}}^p_{d,w}:={{\mathcal {H}}^{p}_{w}}\cap {{\mathcal {D}}}_d. \end{aligned}$$

One of our main results is the following.

Theorem 1

Let $T_g$ be the operator defined by (1.2) for some Dirichlet series g in ${{\mathcal {D}}}$.

• (a) If $g(s)= \sum ^{+\infty }_{n=2}b_n n^{-s}$ is in ${{\mathcal {D}}}_{d}\cap \text {Bloch}(\mathbb {C}_{0})$, then $T_g$ is bounded on ${{\mathcal {H}}}^2_w$ and

  $$\begin{aligned} \left\| T_g\right\| _{{{\mathcal {L}}}({{\mathcal {H}}}_w)}\lesssim \left\| g\right\| _{\text {Bloch}(\mathbb {C}_{0})}. \end{aligned}$$

• (b) If g is in $BMOA(\mathbb {C}_{0})$, then $T_g$ is bounded on ${{\mathcal {H}}}^2_w$ and

  $$\begin{aligned} \left\| T_g\right\| _{{{\mathcal {L}}}({{\mathcal {H}}}_w)}\lesssim \left\| g\right\| _{BMOA(\mathbb {C}_0)}. \end{aligned}$$

• (c) If $T_g$ is bounded on ${{\mathcal {H}}}^2_w$, then g is in $\text {Bloch}(\mathbb {C}_{1/2})$ and

  $$\begin{aligned} \left\| g\right\| _{\text {Bloch}(\mathbb {C}_{1/2})} \lesssim \left\| T_g\right\| _{{{\mathcal {L}}}({{\mathcal {H}}}_w)}. \end{aligned}$$

Via the Bohr lift, ${{\mathcal {H}}^{2}_{w}}$ are $L^2$-spaces of functions on the polydisk $\mathbb {D}^{\infty }$. Precisely, there exists a probability measure $\mu _w$ on $\mathbb {D}^{\infty }$ such that

$$\begin{aligned} \left\| f\right\| ^2_{{{\mathcal {H}}^{2}_{w}}}=\int _{\mathbb {D}^{\infty }} \left| {{\mathcal {B}}}f(z)\right| ^2 d\mu _w(z). \end{aligned}$$

Analogously to the spaces ${{\mathcal {H}}}^p$, we define the space ${{\mathcal {H}}}^p_{w}$, $0<p<\infty$ (see Sect. 2), as the closure of Dirichlet polynomials under the norm (quasi-norm if $0<p<1$)

$$\begin{aligned} \left\| f\right\| _{{{\mathcal {H}}}^p_{w}}= \left\| {{\mathcal {B}}}f\right\| _{L^p\left( \mathbb {D}^{\infty },\mu _w \right) }. \end{aligned}$$

Let ${{\mathcal {X}}_{w}}={{\mathcal {X}}}({{\mathcal {H}}}^2_w)$ be the space of symbols g giving rise to bounded operators $T_g$ on ${{\mathcal {H}}}^2_w$. Our study provides the following strict inclusions:

$$\begin{aligned} BMOA(\mathbb {C}_{0})\cap {{\mathcal {D}}}\subset _{\ne }{{\mathcal {X}}_{w}}\subset _{\ne } \cap _{0<p<\infty }{{\mathcal {H}}}^{p}_{w}. \end{aligned}$$

We will also compare ${{\mathcal {X}}_{w}}$ with other spaces of Dirichlet series, in particular with the dual of ${{\mathcal {H}}}^{1}_{w}$, and the space of symbols g generating a bounded Hankel form

$$\begin{aligned} H_g(fh):=\left\langle fh,g\right\rangle _{{{\mathcal {H}}^{2}_{w}}} \end{aligned}$$

on the weak product ${{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}$. As in the case of ${{\mathcal {H}}}^2$ [13], we only get partial results.

For Dirichlet series involving d primes, we have

$$\begin{aligned} {{\mathcal {D}}}_{d}\cap \text {Bloch}(\mathbb {C}_0)\subset {{\mathcal {D}}}_{d}\cap {{\mathcal {X}}_{w}}\subset _{\ne }{{\mathcal {B}}}^{-1}\text {Bloch}(\mathbb {D}^d). \end{aligned}$$

The paper is organized as follows. Section 2 starts by presenting some properties of the spaces ${{\mathcal {H}}^{2}_{w}}$. As a space of analytic functions on the half-plane $\mathbb {C}_{1/2}$, ${{\mathcal {H}}^{2}_{w}}$ is continuously embedded in a space of Bergman type of $\mathbb {C}_{1/2}$. In view of the Bohr lift, the norm of ${{\mathcal {H}}^{2}_{w}}$ can be expressed in terms of a probability measure $\mu _{w}$ on the polydisk. For $0<p<\infty$, we consider the Bohr–Bergman space ${{\mathcal {H}}}^p_{w}$, and derive equivalent norms for these spaces.

In Sect. 3, we present some properties of the Dirichlet series which belong to a BMO or Bloch space of some half-plane $\mathbb {C}_{\theta }$. In particular, we relate the Carleson measures for both spaces of Dirichlet series and Bergman type spaces.

Section 4 is devoted to the proof of Theorem 1. First we consider the case when g is a function of $p^{-s}_{1}, \ldots ,p^{-s}_{d}$. To prove (b), we observe that the boundedness of $T_g$ on ${{\mathcal {H}}}^2$ implies the boundedness of $T_g$ on ${{\mathcal {H}}}^2_w$. On another hand, combining the fact that ${{\mathcal {H}}^{2}_{w}}$ is embedded in a Bergman type space of the half-plane $\mathbb {C}_{1/2}$ with some characterizations of Carleson measures, we establish that

$$\begin{aligned} {{\mathcal {X}}_{w}}\subset \text {Bloch}(\mathbb {C}_{1/2}). \end{aligned}$$

Compactness and Schatten classes are considered in Sects. 5 and 6.

In Sect. 7, we consider some specific symbols: fractional primitives of translates of a “weighted zeta”-function and homogeneous symbols. These examples will be used in Sect. 8.

In Sect. 8, we investigate the relationship between the boundedness of the Volterra operator $T_g$, the boundedness of the Hankel form

$$\begin{aligned} H_g(fh)=\left\langle fh,g\right\rangle _{{{\mathcal {H}}^{2}_{w}}}, \end{aligned}$$

and the membership of g in the dual of ${{\mathcal {H}}}^1_w$. In particular, we study examples of Hankel forms on Bergman spaces of Dirichlet series, which are the counterparts of the Hilbert multiplicative matrix [12].

Additionally, we show the strictness of the inclusions derived previously

$$\begin{aligned} BMOA(\mathbb {C}_0)\cap {{\mathcal {D}}}\subset _{\ne } {{\mathcal {X}}_{w}}\subset _{\ne } \cap _{0<p<\infty }{{\mathcal {H}}}^{p}_{w}, \end{aligned}$$

and compare the space ${{\mathcal {D}}}_{d}\cap {{\mathcal {X}}_{w}}$ with Bloch spaces.

For two functions f, g, the notation $f=O(g)$ or $f \lesssim g$, means that there exists a constant C such that $f\le C g$ . If $f=O(g)$ and $g=O(f)$, we write $f\asymp g$.

2 The Bohr–Bergman spaces ${{\mathcal {B}}^{2}_{\beta }}$, ${{\mathcal {A}}^{2}_{\beta }}$

2.1 The spaces ${{\mathcal {B}}^{2}_{\beta }}$, ${{\mathcal {A}}^{2}_{\beta }}$

These spaces are related to number theory. The number of divisors of the integer n, d(n), is $d(n)= (\kappa _1+1)\cdots (\kappa _d+1)$ when $n=p^{\kappa }$. We consider the following scale of Hilbert spaces

$$\begin{aligned} {{\mathcal {B}}}^2_{\beta }=\left\{ f(s)= \sum ^{+\infty }_{n=1}a_n n^{-s}:\ \left\| f\right\| _{{{\mathcal {B}}}^2_{\beta }}:=\left( \sum ^{n=1}_{+\infty }\frac{\left| a_n\right| ^2}{\left[ d(n)\right] ^{\beta }}\right) ^{\frac{1}{2}}<\infty \right\} ,\ \text { for } \beta > 0 . \end{aligned}$$

The case $\beta =0$ corresponds to the Hardy space ${{\mathcal {H}}}^2$. The reproducing kernels of ${{\mathcal {B}}}^2_{\beta }$ are

$$\begin{aligned} K^{{{\mathcal {B}}^{2}_{\beta }}}(s,u)=\zeta _{\beta }(s+\overline{u}),\text { where } \zeta _{\beta }(s)=\sum _{n=1}^{+\infty }\left[ d(n)\right] ^{\beta }n^{-s}. \end{aligned}$$

It is shown in [30] that there exists $\phi _{\beta }(s)$, an Euler product which converges absolutely in $\mathbb {C}_{1/2}$, such that

$$\begin{aligned} \zeta _{\beta }(s)=\left[ \zeta (s)\right] ^{2^{\beta }}\phi _{\beta }(s),\text { and }\phi _{\beta }(1)\ne 0. \end{aligned}$$

Another family of spaces arises from the so-called generalized divisor function. For $\gamma >0$, the numbers $d_{\gamma }(n)$ are defined by the relation

$$\begin{aligned} \zeta ^{\gamma }(s)=\sum ^{+\infty }_{n=1}d_{\gamma }(n)n^{-s} . \end{aligned}$$

A computation involving Euler products shows that we have

$$\begin{aligned} d_{\gamma }(p^r)=\frac{\gamma (\gamma +1)\cdots (\gamma +r-1) }{r!},\ \text { for }p\in \mathbb {P},\ \text { and any integer }r. \end{aligned}$$

From its definition, $d_{\gamma }$ is a multiplicative function, i.e. $d_{\gamma }(kl)=d_{\gamma }(k)d_{\gamma }(l)$ if k and l are relatively prime. Thus, $d_{\gamma }(n)$ can be computed explicitly from the decomposition $n=p^{\kappa }$.

We define the spaces

$$\begin{aligned} {{\mathcal {A}}}^2_{\beta }=\left\{ f(s)= \sum ^{+\infty }_{n=1}a_n n^{-s}:\ \left\| f\right\| _{{{\mathcal {A}}}^2_{\beta }}:=\left( \sum ^{n=1}_{+\infty }\frac{\left| a_n\right| ^2}{d_{\beta +1}(n)}\right) ^{\frac{1}{2}}<\infty \right\} ,\ \text { for }\beta >0, \end{aligned}$$

with reproducing kernels $K^{{{\mathcal {A}}^{2}_{\beta }}}(s,u)=\zeta ^{\beta +1}(s+\overline{u})$.

Notice that, in each case, the reproducing kernel has the form

$$\begin{aligned} K^{{{\mathcal {H}}^{2}_{w}}}(s,u)=Z_w(s+{\overline{u}}), \end{aligned}$$

where $Z_w(s):=\sum ^{+\infty }_{n=1}w_n n^{-s}$ has a singularity at $s=1$, with an estimate of the type

$$\begin{aligned} Z_w(s)=C_w(s-1)^{-\left( \delta +1\right) }\left[ 1+O(1)\right] . \end{aligned}$$

(2.1)

2.2 Bohr–Bergman spaces on $\mathbb {D}^\infty$

The Bohr correspondence is an isometry between ${{\mathcal {H}}}^2_{w}$ and the weighted Bergman space of the infinite polydisk

$$\begin{aligned} H^2_w(\mathbb {D}^{\infty })= \left\{ \sum _{\nu \in \mathbb {N}^{\infty }_{0,\text {fin}}}a_{\nu } z^{\nu }:\ \sum _{\nu }\frac{\left| a_{\nu }\right| ^2}{w_{\nu }}<\infty \right\} ,\ \text { where } w_{\nu }=\prod _{j}w_{\nu _j}. \end{aligned}$$

In particular, the space ${{\mathcal {H}}}^2$ is identified with the Hardy space $H^2(\mathbb {T}^{\infty })$ [19].

Let us consider the following probability measures on the unit disk $\mathbb {D}$,

$$\begin{aligned} dm_{w}(z):= & {} M(\left| z\right| ^2)dV(z),\\ \text { where }M(r)= & {} {\left\{ \begin{array}{ll}\frac{1}{\Gamma (\beta )}\left( \log \frac{1}{r}\right) ^{\beta -1},\ &{} \text { if }w_n=\left[ d(n)\right] ^{\beta } ,\\ \beta (1-r)^{\beta -1},\ &{} \text { if } w_n= d_{\beta +1}(n) \end{array}\right. }\ \beta >0. \end{aligned}$$

On the finite polydisk $\mathbb {D}^d$$(d\in \mathbb {N})$, the corresponding Bergman spaces $H^2_w(\mathbb {D}^d)$ - specifically $B^{2}_{\beta }(\mathbb {D}^d)$ and $A^{2}_{\beta }(\mathbb {D}^d)$- are the $L^2-$closures of polynomials with respect to the norm

$$\begin{aligned} \left\| f\right\| _{H^2_w(\mathbb {D}^d)} :=\left( \int _{\mathbb {D}^d}\left| f(z_1,\ldots ,z_d)\right| ^2 dm_{w}(z_1)\times \cdots \times dm_{w}(z_d)\right) ^{1/2} \end{aligned}$$

If $f(z)=\sum _{n\in \mathbb {N}^d}a_n z^n$ is defined on $\mathbb {D}^d$, we have

$$\begin{aligned} \left\| f\right\| ^{2}_{B^{2}_{\beta }(\mathbb {D})}= & {} \sum _{n\in \mathbb {N}}\frac{\left| a_n\right| ^2}{\left( n+1\right) ^{\beta }}\nonumber \\ \text { and }\left\| f\right\| ^{2}_{A^{2}_{\beta }(\mathbb {D})}= & {} \sum _{n\in \mathbb {N}} \left| a_n\right| ^2\frac{n!}{(\beta +1)(\beta +2)\cdots (\beta +n)}. \end{aligned}$$

(2.2)

When d is finite, the estimate

$$\begin{aligned} \frac{n!}{(\beta +1)(\beta +2)\cdots (\beta +n)}\asymp (1+n)^{-\beta } \end{aligned}$$

yields that, the spaces $B^{2}_{\beta }(\mathbb {D}^d)$ and $A^{2}_{\beta }(\mathbb {D}^d)$ coincide as sets, with equivalent norms. However, the norms are no longer equivalent in the case of infinitely many variables.

The ${{\mathcal {H}}^{2}_{w}}$-norm will be computed via the rotation invariant probability measure

$$\begin{aligned} d\mu _{w}(\chi )=dm_{w}(\chi _1)\times dm_{w}(\chi _2) \times dm_{w}(\chi _3) \times \cdots \text { on }\mathbb {D}^{\infty }. \end{aligned}$$

Applying the Bohr lift to a Dirichlet series $f(s)= \sum ^{+\infty }_{n=1}a_n n^{-s}$, and using (2.2) for each variable, one obtains the following formula (see [5] in the case of ${{\mathcal {B}}^{2}_{\beta }}$)

$$\begin{aligned} \int _{\mathbb {D}^{\infty }}\left| {{\mathcal {B}}}f(\chi )\right| ^2 d\mu _{w}(\chi )= \sum ^{+\infty }_{n=1}\frac{\left| a_n\right| ^2}{w_n}=\left\| f\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}. \end{aligned}$$

Definition 2

For $0< p<\infty$, the Bohr–Bergman spaces of Dirichlet series ${{\mathcal {B}}}^{p}_{\beta }$ and ${{\mathcal {A}}}^{p}_{\beta }$ - denoted by ${{\mathcal {H}}}^{p}_{w}$ - are the completions of the Dirichlet polynomials in the norm (quasi norm when $0<p<1$)

$$\begin{aligned} \left\| f\right\| ^{p}_{{{\mathcal {H}}}^{p}_{w}}:= \int _{\mathbb {D}^{\infty }}\left| {{\mathcal {B}}}f(\chi )\right| ^p d\mu _{w}(\chi ). \end{aligned}$$

The Kronecker flow of the point $\chi =(\chi _1,\chi _2,\ldots )\in \mathbb C^{\infty }$ is given by

$$\begin{aligned} {{\mathcal {T}}}_t(\chi )=\left( 2^{-it }\chi _1, 3^{-it }\chi _2,5^{-it }\chi _3,\ldots \right) ,\ t\in \mathbb R, \end{aligned}$$

which defines an ergodic flow on $\mathbb {T}^{\infty }$ by Kronecker’s theorem.

Therefore, it follows from Fubini’s Theorem that, for any rotation invariant probability measure $d\nu$ on $\mathbb {D}^{\infty }$ and any probability measure $d\lambda$ on $\mathbb {R}$, we have

$$\begin{aligned} \left\| f\right\| ^{p}_{L^{p}\left( \mathbb {D}^{\infty },d\nu \right) }=\int _{\mathbb {D}^{\infty }}\int _{\mathbb {R}}\left| \left( {{\mathcal {B}}}f\right) ({{\mathcal {T}}}_t\chi )\right| ^p d\lambda (t)d\nu \left( \chi \right) . \end{aligned}$$

(2.3)

2.3 On the half-plane $\mathbb {C}_{1/2}$

For $\theta \in \mathbb {R}$, let $\tau _{\theta }$ be the following mapping from $\mathbb {D}$ to $\mathbb {C}_{\theta }$,

$$\begin{aligned} \tau _{\theta }(z)=\theta +\frac{1+z}{1-z}. \end{aligned}$$

(2.4)

For $\delta >0$, the conformally invariant Bergman space $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$ is the space of those functions f which are analytic in $\mathbb {C}_{1/2}$, and such that

$$\begin{aligned} \left\| f\right\| ^{2}_{A_{i,\delta }\left( \mathbb {C}_{1/2}\right) }:=\left\| f\circ \tau _{1/2} \right\| ^{2}_{A^2_{\delta }\left( \mathbb {D}\right) }=4^{\delta }\delta \int _{\mathbb {C}_{1/2}}\left| f(s)\right| ^2 \frac{\left( \sigma -\frac{1}{2}\right) ^{\delta -1}}{\left| s+\frac{1}{2}\right| ^{2\delta +2}}dm(s)<\infty . \end{aligned}$$

The weights w of the class ${{\mathcal {W}}}$ satisfy a Chebyshev-type estimate

$$\begin{aligned} \sum _{n\le x}w_n \asymp x\left( \log x\right) ^{\delta },\ \text { where } \delta =\delta (w):= {\left\{ \begin{array}{ll} 2^{\beta }-1 &{}\text { if }w_n=\left[ d(n)\right] ^{\beta },\\ \beta &{}\text { if }w_n=d_{\beta +1}(n). \end{array}\right. }\nonumber \\ \end{aligned}$$

(2.5)

For any real number $\tau$, set $S_{\tau }=\left[ \frac{1}{2},1\right] \times \left[ \tau ,\tau +1\right]$. As mentioned in the introduction, the Dirichlet series which belong the ${{\mathcal {H}}}^{2}_{w}$ absolutely converge in $\mathbb {C}_{1/2}$. The space ${{\mathcal {H}}^{2}_{w}}$ is locally embedded in $A_{i,\delta (w)}\left( \mathbb {C}_{1/2}\right)$ [23, 25], which means

$$\begin{aligned} \sup _{\tau \in \mathbb {R}}\int _{S_{\tau }}\left| f(s)\right| ^2\frac{\left( \sigma -\frac{1}{2}\right) ^{\delta -1}}{\left| s+\frac{1}{2}\right| ^{2\delta +2}}dm(s)\le c\left( {{\mathcal {H}}^{2}_{w}}\right) \left\| f\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}. \end{aligned}$$

Since functions in ${{\mathcal {H}}}^2_w$ are uniformly bounded in $\mathbb {C}_1$, these embeddings are global (see [5, 8]).

Lemma 1

Let $\delta =\delta (w)$ be defined in (2.5). Then ${{\mathcal {H}}^{2}_{w}}$ is continuously embedded in $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$.

2.4 Generalized vertical limits

Every $\chi =(\chi _1,\chi _2,\ldots )$ in $\mathbb C^{\infty }$ defines a completely multiplicative function by the formula $\chi (n)= \chi ^{\kappa }$, where $n=p^{\kappa }$. For f of the form (1.1), the twisted Dirichlet series [5, 6], is defined by

$$\begin{aligned} f_{\chi }(s)= \sum ^{+\infty }_{n=1}a_n\chi (n) n^{-s}. \end{aligned}$$

(2.6)

Notice that if $\chi \in \mathbb {T}^{\infty }$, $f_{\chi }$ is the vertical limit of f, introduced in [19].

We also consider the translations $f_{\delta }(s)=f(s+\delta )$, $\delta \in \mathbb {R}$. For those $\chi \in \mathbb {D}^{\infty }$ and $s=\sigma +it$ for which the series (2.6) converges, we have

$$\begin{aligned} f_{\chi }(s)= \left( {{\mathcal {B}}}f_{\sigma }{{\mathcal {T}}}_t\right) (\chi ). \end{aligned}$$

(2.7)

When f is in ${{\mathcal {H}}^{2}_{w}}$, the Cauchy-Schwarz inequality implies that (2.7) holds whenever $s\in \mathbb {C}_{1/2}$ and $\chi \in \overline{\mathbb {D}}^{\infty }$. By the Rademacher-Menchov Theorem (see [22]), (2.7) can be extended in the following way (the argument given in [5] for ${{\mathcal {B}}}^2_{\beta }$ remains true for ${{\mathcal {A}}}^2_{\beta }$).

Lemma 2

If f is in ${{\mathcal {H}}^{2}_{w}}$, the Dirichlet series $f_{\chi }$ as defined in (2.6) converges in $\mathbb {C}_{0}$ for almost every $\chi \in \mathbb {D}^{\infty }$, with respect to $\mu _{w}$.

Recall that $\tau _{\theta }$, $\theta \in \mathbb {R}$, is the conformal mapping defined in (2.4). For $0< p<\infty$, the conformally invariant Hardy space $H^{p}_{i}\left( \mathbb {C}_{\theta }\right)$, is the space of those functions f such that $f\circ \tau _{\theta }$ is in $H^p(\mathbb {T})$, the usual Hardy space of the unit disk. Setting $d\lambda (t)=\pi ^{-1}(1+t^2)^{-1}dt$, we get

$$\begin{aligned} \left\| f\right\| ^p_{H^{p}_{i}\left( \mathbb {C}_{\theta }\right) }= \int _{\mathbb {R}}\left| f\left( \theta +it\right) \right| ^{p}d\lambda (t)=\frac{1}{2\pi }\int ^{\pi }_{-\pi }\left| f\circ \tau _{\theta }(u) \right| ^p du, \ \text { for } f\in H^{p}_{i}\left( \mathbb {C}_{\theta }\right) . \end{aligned}$$

Let f be in ${{\mathcal {H}}}^p_{w}$. In view of relation (2.3), and using the same argument as in [6, 19], one can prove that for almost all $\chi$, with respect to $\mu _{w}$ , $f_{\chi }$ can be extended analytically on $\mathbb {C}_0$ to an element of $H^{p}_{i}\left( \mathbb {C}_{0}\right)$.The norm of f in ${{\mathcal {H}}}^p_{w}$ can be expressed as

$$\begin{aligned} \left\| f\right\| ^{p}_{{{\mathcal {H}}}^p_{w}}=\int _{\mathbb {D}^{\infty }}\left\| f_{\chi }\right\| ^{p}_{H^{p}_{i}\left( \mathbb {C}_{0}\right) }d\mu _{w}(\chi ). \end{aligned}$$

(2.8)

2.5 A Littlewood–Paley formula

We now derive another expression for the norm in ${{\mathcal {H}}}^p_{w}$.

Proposition 1

Let $\lambda$ be a probability measure on $\mathbb {R}$, and $p\ge 1$.

• (a) If $f\in {{\mathcal {H}}^{p}_{w}}$, then $\left\| f\right\| ^{p}_{{{\mathcal {H}}^{p}_{w}}}\asymp I_p(f)$, where

  $$\begin{aligned} I_p(f):= & {} \left| f(+\infty )\right| ^p\\&+4\int _{\mathbb {D}^{\infty }}\int _{\mathbb {R}}\int ^{+\infty }_{0}\left| f_{\chi }( y+it)\right| ^{p-2}\left| f'_{\chi }( y+it)\right| ^2ydy d\lambda (t) d\mu _{w}(\chi ). \end{aligned}$$

  When $p=2$, we have $\left\| f\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}= I_2(f)$.

• (b) Let $f\in {{\mathcal {D}}}$, $f(s)=\sum ^{+\infty }_{n=1}a_n n^{-s},$ such that f and $f_{\chi }$ converge on $\mathbb {C}_0$ for a.a. $\chi \in \mathbb {D}^{\infty }$. If $I_p(f)<\infty$, then $f\in {{\mathcal {H}}^{p}_{w}}$.

Proof

Since the real variable t corresponds to a rotation in each variable of $\mathbb {D}^{\infty }$, the rotation invariance of $\mu _w$ entails that $I_p(f)$ does not depend on the choice of the probability measure $\lambda$. For general $p\ge 1$, we prove (a), by using (2.8). We adapt the argument from [10] (for ${{\mathcal {H}}}^p$), by integrating over the polydisk $\mathbb {D}^{\infty }$ instead of the polytorus $\mathbb {T}^{\infty }$.

Suppose f is in ${{\mathcal {H}}^{2}_{w}}$, and take $y>0$. From (2.3) and the rotation invariance, we obtain

$$\begin{aligned} \int _{\mathbb {R}}\int _{\mathbb {D}^\infty }\left| f'_{ \chi }(y+it)\right| ^2 d\mu _{w}( \chi ) d\lambda (t)&=\int _{\mathbb {D}^\infty }\left| {{\mathcal {B}}}f'_{ y}(\chi )\right| ^2 d\mu _{w}( \chi )\\&=\sum ^{+\infty }_{n=1}\frac{\left| a_n\right| ^2}{w_n}(\log n)^2 n^{-2y}. \end{aligned}$$

Integration against y on $(0,+\infty )$ gives the formula (see details in [7] for the case of ${{\mathcal {H}}}^2$).

If f is as in (b), the integrand in $I_p(f)$ is measurable. For $\chi \in \mathbb {D}^{\infty }$, the change of variables $s=y+it=\omega (z)=2\frac{1+z}{1-z}$ transfers the Littlewood–Paley formula from $\mathbb {D}$ to $\mathbb {C}_0$,

$$\begin{aligned}&\int _{\mathbb {R}}\left| f_{ \chi }(it)\right| ^p \frac{2}{\pi (2^2+t^2)}dt\\&\quad \asymp \left| f_{ \chi }(2)\right| ^p\\&\qquad +\int _{\mathbb {D}}\left( 1-\left| z\right| ^2\right) \left| f_{ \chi }(\omega (z))\right| ^{p-2}\left| f'_{ \chi }(\omega (z))\right| ^2\left| \omega '(z)\right| ^2 dV(z) \\&\quad \asymp \left| f_{ \chi }(2)\right| ^p\\&\qquad +\int ^{+\infty }_{0}\int _{\mathbb {R}}\frac{2y}{(y+2)^2+t^2}\left| f_{ \chi }(y+it)\right| ^{p-2}\left| f'_{ \chi }(y+it)\right| ^2dtdy\\&\quad \lesssim \left\| f^*\right\| ^{p}_{L^{\infty }(\overline{\mathbb {C}_2})}\\&\qquad +\int ^{+\infty }_{0}\int _{\mathbb {R}}\frac{y}{1+t^2}\left| f_{ \chi }(y+it)\right| ^{p-2}\left| f'_{ \chi }(y+it)\right| ^2dtdy, \end{aligned}$$

where $f^*(s):=\sum ^{+\infty }_{n=1}\left| a_n\right| n^{-s}$ is bounded on $\overline{\mathbb {C}_2}$.

Integrating on $\mathbb {D}^\infty$ with respect to $\mu _w$, and using (2.3), we get that

$$\begin{aligned} \left\| {{\mathcal {B}}}f\right\| ^{p}_{L^p(\mathbb {D}^\infty , \mu _w)} \lesssim \left\| f^*\right\| ^{p}_{L^{\infty }(\overline{\mathbb {C}_2})}+I_p(f)<\infty . \end{aligned}$$

Therefore, ${{\mathcal {B}}}f\in L^p(\mathbb {D}^\infty , \mu _w)$. The martingale $(A_m {{\mathcal {B}}}f)_m$ (with respect to the increasing sequence of $\sigma$-algebras of the sets $\mathbb {D}^m\times \left\{ 0\right\}$) converges in $L^p(\mathbb {D}^\infty , \mu _w)$ to ${{\mathcal {B}}}f$. Polynomial approximation in the Bergman spaces of the finite polydisks $\mathbb {D}^m$ shows that ${{\mathcal {B}}}f$ is in ${{\mathcal {B}}}{{\mathcal {H}}^{p}_{w}}$.

$\square$

3 Spaces of symbols of Volterra operators in half-planes

If g is in ${{\mathcal {D}}}$, the definition (1.2) of $T_g$ shows that we can assume that $g\left( +\infty \right) =0$, i.e.

$$\begin{aligned} g(s)=\sum ^{+\infty }_{n=2}b_n n^{-s}. \end{aligned}$$

As in the study of Volterra operators on Bergman spaces the unit disk [2], and on the space of Dirichlet series ${{\mathcal {H}}}^2$ [13], the boundedness of $T_g$ on ${{\mathcal {H}}^{2}_{w}}$ will be related to Carleson measures, and to the membership of g to a BMO space or a Bloch space.

Let Y be either ${{\mathcal {H}}^{2}_{w}}$ or the Bergman space $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$, $\delta >0$. A positive Borel measure $\mu$ on $\mathbb {C}_{1/2}$ is called a Carleson measure for Y if there exists a constant C such that,

$$\begin{aligned} \int _{ \mathbb {C}_{1/2} } \left| f\right| ^2 d\mu \le C\left\| f\right\| ^2_{Y}\text { for all }f\in Y . \end{aligned}$$

The smallest such constant, denoted by $\left\| \mu \right\| _{CM(Y)}$, is called the Carleson constant for $\mu$ with respect to Y. A Carleson measure $\mu$ is a vanishing Carleson measure for Y if we have

$$\begin{aligned} \lim _{k\rightarrow \infty }\int _{ \mathbb {C}_{1/2} } \left| f_k\right| ^2 d\mu =0, \end{aligned}$$

for every weakly compact sequence $(f_k)_k$ in Y (which means that $\left\| f_k\right\| _{Y}$ is bounded and $f_k(s)\rightarrow 0$ on every compact set of $\mathbb {C}_{1/2}$).

3.1 BMO spaces of Dirichlet series

The space $BMOA(\mathbb {C}_{\theta })$ consists of holomorphic functions g in the half-plane $\mathbb {C}_{\theta }$ which satisfy

$$\begin{aligned} \left\| g\right\| _{BMO(\mathbb {C}_{\theta })}:=\sup _{I\subset \mathbb {R}}\frac{1}{\left| I\right| }\int _{I}\left| g(\theta +it)-\frac{1}{\left| I\right| }\int _{I}g(\theta +i\tau )d\tau \right| dt<\infty . \end{aligned}$$

Any g in ${{\mathcal {D}}}\cap BMOA(\mathbb {C}_{0})$ has an abscissa of boundedness $\sigma _b\le 0$ (Lemma 2.1 of [13]).

The space $VMOA(\mathbb {C}_0)$ consists in those functions g in $BMOA(\mathbb {C}_0)$ such that

$$\begin{aligned} \lim _{\delta \rightarrow 0^+}\sup _{\left| I\right| <\delta }\frac{1}{\left| I\right| }\int _{I}\left| f(it) - \frac{1}{\left| I\right| }\int _I f(i\tau ) d\tau \right| dt=0. \end{aligned}$$

3.2 Bloch spaces of Dirichlet series

The Bloch space $\text {Bloch}(\mathbb {C}_{\theta })$ consists of holomorphic functions in the half-plane $\mathbb {C}_{\theta }$ which satisfy

$$\begin{aligned} \left\| g\right\| _{\text {Bloch}(\mathbb {C}_{\theta })}:=\sup _{\sigma +it\in \mathbb {C}_{\theta }}\left( \sigma -\theta \right) \left| f'(\sigma +it)\right| . \end{aligned}$$

Lemma 3

If g be in ${{\mathcal {D}}}\cap \text {Bloch}(\mathbb {C}_{0})$.

• (a) Its abscissa of boundedness satifies $\sigma _b\le 0$.

• (b) For every $\chi \in \mathbb {D}^\infty$, $g_{\chi }$ is in $\text {Bloch}(\mathbb {C}_0)$, and $\left\| g_{\chi }\right\| _{\text {Bloch}(\mathbb {C}_0)}\le \left\| g\right\| _{\text {Bloch}(\mathbb {C}_0)}$.

• (c) Suppose that $y_0>\frac{1}{2}$. Then there exists a constant $C=C(y_0)$, such that,

  $$\begin{aligned} \left| g'_{\chi }(y+it)\right| \le C 2^{-y} \left\| g\right\| _{\text {Bloch}(\mathbb {C}_0)} ,\ \text { for all }\chi \in \mathbb {D}^\infty ,\ t\in \mathbb {R}, \ y\ge y_0. \end{aligned}$$

Proof

Let $\epsilon >0$. If $s=\sigma +it$ is in $\mathbb {C}_0$, the definition of the Bloch-norm implies that

$$\begin{aligned} \epsilon \left| g'(\epsilon +s)\right| \le (\epsilon +\sigma )\left| g'(\epsilon +s)\right| \le \left\| g\right\| _{\text {Bloch}(\mathbb {C}_{0})}. \end{aligned}$$

It follows that $g'$, and then g is bounded in $\mathbb {C}_{\epsilon }$; (a) is proved.

Now fix $\sigma >0$. Let $m\ge 1$ be an integer, and $z=(z_1,\ldots ,z_m,z_{m+1},\ldots ),$$\chi$ in $\mathbb {D}^\infty$. From the properties of ${{\mathcal {H}}}^\infty$ and the proof of (a), we have

$$\begin{aligned} \left| A_m {{\mathcal {B}}}(g'_{\sigma })_{\chi }(z)\right| =\left| A_m {{\mathcal {B}}}g'_{\sigma }(z.{\chi })\right| \le \left\| {{\mathcal {B}}}g'_{\sigma }\right\| _{H^\infty (\mathbb {T}^\infty )}=\left\| g'_{\sigma }\right\| _{{{\mathcal {H}}}^\infty }, \end{aligned}$$

and $\left\| (g'_{\sigma })_{\chi }\right\| _{{{\mathcal {H}}}^\infty }=\left\| {{\mathcal {B}}}(g'_{\sigma })_{\chi }\right\| _{H^\infty (\mathbb {T}^\infty )}\le \left\| g'_{\sigma }\right\| _{{{\mathcal {H}}}^\infty }$. Therefore, $(g'_\sigma )_{\chi }$ is in ${{\mathcal {H}}}^\infty$; (b) holds, due to

$$\begin{aligned} \sigma \left| g'_{\chi }(\sigma +it)\right| \le \left\| g\right\| _{\text {Bloch}(\mathbb {C}_{0})},\ \text { for all }t\in \mathbb {R}, \chi \in \mathbb {T}^\infty ,\sigma >0. \end{aligned}$$

If $0<\delta <y_0-\frac{1}{2}$, the Cauchy-Schwarz inequality and Parseval’s relation induce that

$$\begin{aligned} \left| g'_{\chi }(y+it)\right| ^2&\le \left( \sum ^{+\infty }_{n=2}\left| b_n\right| (\log n) n^{-y}\right) ^2 = \left( \sum ^{+\infty }_{n=2}\left| b_n\right| (\log n) n^{-\frac{\delta }{2}} n^{-\left( \frac{\delta }{2}+\frac{1}{2}\right) }n^{-\left( y-\frac{1}{2}-\delta \right) }\right) ^2\\&\lesssim \zeta (1+\delta )2^{-2y}\left\| {{\mathcal {B}}}g'_{\delta /2}\right\| ^{2}_{H^{2}\left( \mathbb {T}^\infty \right) }. \end{aligned}$$

We now get (c) from the chain of inequalities

$$\begin{aligned} \left\| {{\mathcal {B}}}g'_{\delta /2}\right\| _{H^{2}\left( \mathbb {T}^\infty \right) }\le \left\| {{\mathcal {B}}}g'_{\delta /2}\right\| _{H^{\infty }\left( \mathbb {T}^\infty \right) }=\left\| g'_{\delta /2}\right\| _{{{\mathcal {H}}}^{\infty }}\le \frac{2}{\delta }\left\| g\right\| _{\text {Bloch}(\mathbb {C}_{0})}, \end{aligned}$$

$\square$

Now, recall several characterizations of Bloch functions, which are extracted from [2, 18].

Lemma 4

Assume $\delta >0$. For g holomorphic in $\mathbb {C}_{\theta }$, the following are equivalent:

• (a) $g\in \text {Bloch}(\mathbb {C}_{\theta })$;

• (b) $h=g\circ \tau _{\theta }\in \text {Bloch}(\mathbb {D})$;

• (c) The measure $d\mu _{\mathbb {C}_{\theta }, g}(s)=\left| g'(\sigma +it)\right| ^2\frac{\left( \sigma -\theta \right) ^{\delta +1}}{\left| s-\theta +1\right| ^{2\delta +2}}d\sigma dt$ is a Carleson measure for $A_{i,\delta }(\mathbb {C}_{\theta })$;

• (d) The measure $d\mu _{\mathbb {D},h}(z)=\left| h'(z)\right| ^2 \left( 1-\left| z\right| ^2\right) ^{\delta +1}dm_1(z)$ is a Carleson measure for $A^2_{\delta }(\mathbb {D})$;

• (e) The operator $J_h$, given by

  $$\begin{aligned} J_hf(z)=\int ^{z}_{0}f(t) h'(t)dt, \end{aligned}$$

  is bounded on $A^2_{\delta }(\mathbb {D})$.

Moreover, the quantities

$$\begin{aligned} \left\| g\right\| _{\text {Bloch}(\mathbb {C}_{\theta })}, \ \left\| \mu _{\mathbb {C}_{\theta }, g}\right\| _{CM(\mathbb {C}_{\theta })}, \left\| J_g\right\| _{{{\mathcal {L}}}\left( A^2_{\delta }(\mathbb {D})\right) } \end{aligned}$$

are comparable.

The little Bloch space is the space

$$\begin{aligned} \text {Bloch}_0(\mathbb {C}_{\theta })=\left\{ f\in \text {Bloch}(\mathbb {C}_{\theta })\ : \ \lim _{\sigma \rightarrow \theta }\left( \sigma -\theta \right) \left| g'(s)\right| =0\right\} . \end{aligned}$$

The membership in $\text {Bloch}_0(\mathbb {C}_{\theta })$ is characterized by a little oh version of Lemma 4, involving vanishing Carleson measures.

We show that Dirichlet polynomials are dense in ${{\mathcal {D}}}\cap \text {Bloch}_0(\mathbb {C}_{0})$. For $g(s)=\sum _{n\ge 1}b_n n^{-s}$, the partial sum operator is defined by $S_N g(s)=\sum _{n= 1}^{N}b_n n^{-s}$.

Proposition 2

Let g be in $\text {Bloch}_0(\mathbb {C}_{0})\cap {{\mathcal {D}}}$, and $\epsilon >0$. Then there exists P in ${{\mathcal {P}}}$ such that

$$\begin{aligned} \left\| g-P\right\| _{\text {Bloch}(\mathbb {C}_{0})}\le \epsilon . \end{aligned}$$

If in addition g is in ${{\mathcal {D}}}_d$, P can be chosen in ${{\mathcal {D}}}_d$.

Proof

For every $\delta >0$, $g_\delta =g(\delta +.)$ is also in $\text {Bloch}_0(\mathbb {C}_{0})$. As $\delta$ tends to 0, $(g_\delta )_\delta$ converges to g uniformly on compact sets of $\mathbb {C}_0$, and $\lim _{\sigma \rightarrow 0^+}\sigma \left| g'_\delta (s)\right| =0,$ uniformly with respect to $\delta \in (0,1)$. It then follows from [3] that $\lim _{\delta \rightarrow 0^+}\left\| g-g_\delta \right\| _{\text {Bloch}(\mathbb {C}_{0})}=0.$ Thus, we can choose $\delta >0$ such that $\left\| g-g_\delta \right\| _{\text {Bloch}(\mathbb {C}_{0}) } \le \frac{\epsilon }{2}.$ Since $\sigma _b(g)=\sigma _u(g)\le 0$, the partial sums $\left( S_N g\right) _N$ converge uniformly to g in $\overline{\mathbb {C}_\delta }$, $\lim _{N\rightarrow +\infty }\left\| S_N g_\delta -g_\delta \right\| _{{{\mathcal {H}}}^{\infty }}=0.$ For large N, the triangle inequality implies that

$$\begin{aligned} \left\| g-S_N g_\delta \right\| _{\text {Bloch}(\mathbb {C}_{0})}&\le \left\| g-g_\delta \right\| _{\text {Bloch}(\mathbb {C}_{0}) }+\left\| g_\delta -S_N g_\delta \right\| _{\text {Bloch}(\mathbb {C}_{0})}\\&\le \frac{\epsilon }{2}+2 \left\| S_N g_\delta -g_\delta \right\| _{{{\mathcal {H}}}^{\infty }}\le \epsilon . \end{aligned}$$

$\square$

3.3 Carleson measures on the half-plane $\mathbb {C}_{1/2}$

On $\mathbb {C}_{1/2}$, we consider Carleson squares

$$\begin{aligned} Q(s_0)=\left( \frac{1}{2}, \sigma _{0}\right] \times \left[ t_0-\frac{\epsilon }{2},t_0+\frac{\epsilon }{2}\right] , \text { where }s_0=\sigma _{0}+it_0\in \mathbb {C}_{1/2} \end{aligned}$$

is the midpoint of the right edge of the square and $\epsilon =\sigma _{0}-\frac{1}{2}.$

We need the following property (see Section 7.2 in [31]).

Lemma 5

Let $\delta >0$ and let $\mu$ be a Borel measure on $\mathbb {C}_{1/2}$. Then $\mu$ is a Carleson measure for $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$ if and only if, for every square $Q(s_0)$, with $s_0=\sigma _{0}+it_0$, we have

$$\begin{aligned} \mu \left( Q(s_0)\right) =O\left( \left( 2\sigma _{0}-1\right) ^{\delta +1}\right) \text { as }\sigma _0\rightarrow \left( \frac{1}{2}\right) ^+. \end{aligned}$$

In addition, $\mu$ is a vanishing Carleson measure for $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$ if and only if, uniformly for $t_0$ in $\mathbb {R}$,

$$\begin{aligned} \mu \left( Q(s_0)\right) =o\left( \left( 2\sigma _{0}-1\right) ^{\delta +1}\right) \text { as }\sigma _0\rightarrow \left( \frac{1}{2}\right) ^+. \end{aligned}$$

By Lemma 1, ${{\mathcal {H}}^{2}_{w}}$ is embedded in the Bergman-type space $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$, the exponent $\delta =\delta (w)$ being defined in (2.5). Bounded Carleson measures for both spaces ${{\mathcal {H}}^{2}_{w}}$ and $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$ have been compared in [8, 23, 24]. We extend their results.

Lemma 6

Let $\mu$ be a positive Borel measure on $\mathbb {C}_{1/2}$.

• (1) If $\mu$ is a Carleson measure (resp. vanishing Carleson measure) for ${{\mathcal {H}}^{2}_{w}}$, then $\mu$ is a Carleson measure (resp. vanishing Carleson measure) for $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$ and

  $$\begin{aligned} \left\| \mu \right\| _{CM\left( A_{i,\delta }\left( \mathbb {C}_{1/2}\right) \right) }\lesssim \left\| \mu \right\| _{CM\left( {{\mathcal {H}}^{2}_{w}}\right) }. \end{aligned}$$

• (2) Assume that $\mu$ has bounded support. If $\mu$ is a Carleson measure (resp. vanishing Carleson measure) for $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$, then $\mu$ is a Carleson measure (resp. vanishing Carleson measure) for ${{\mathcal {H}}^{2}_{w}}$ and

  $$\begin{aligned} \left\| \mu \right\| _{CM\left( {{\mathcal {H}}^{2}_{w}}\right) }\lesssim \left\| \mu \right\| _{CM\left( A_{i,\delta }\left( \mathbb {C}_{1/2}\right) \right) }. \end{aligned}$$

Proof

Suppose that $\mu$ is a Carleson measure for ${{\mathcal {H}}^{2}_{w}}$, and let $Q(s_0)$ be a small Carleson square in $\mathbb {C}_{1/2}$. For the test function $f_{s_0}(s)=K^{{{\mathcal {H}}^{2}_{w}}}(s,s_0)$, we have

$$\begin{aligned} \int _{Q(s_0)}\left| f_{s_0}\right| ^2 d\mu \le \int _{\mathbb {C}_{1/2}}\left| f_{s_0}\right| ^2 d\mu \le C(\mu )\left\| K^{{{\mathcal {H}}^{2}_{w}}}(.,s_0)\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}\lesssim Z_w\left( \mathfrak {R}s_0\right) . \end{aligned}$$

From the estimate of $Z_w$ (2.1) and Lemma 5, $\mu$ is a Carleson measure for $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$, since

$$\begin{aligned} \left( \mathfrak {R}s_0-\frac{1}{2}\right) ^{-2\left( \delta +1\right) }\mu \left( Q(s_0)\right) \lesssim \left( \mathfrak {R}s_0-\frac{1}{2}\right) ^{-\left( \delta +1\right) }. \end{aligned}$$

For $\mu$ a Carleson measure for $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$ with bounded support, (2) holds [23, 24].

As for vanishing Carleson measures, the reasoning used in [8] for ${{\mathcal {B}}^{2}_{\beta }}$ can be transfered to the spaces ${{\mathcal {A}}^{2}_{\beta }}$, with the test functions

$$\begin{aligned} f_k(s)=\frac{K^{{{\mathcal {H}}^{2}_{w}}}(s,s_k)}{\left\| K^{{{\mathcal {H}}^{2}_{w}}}(.,s_k)\right\| _{{{\mathcal {H}}^{2}_{w}}}}, \end{aligned}$$

where $s_k=1/2+\epsilon _k+i \tau _k$ is a sequence in $\mathbb {C}_{1/2}$ such that $\epsilon _k\rightarrow 0$. $\square$

We also require an equivalent norm for $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$, when $\delta >0$. For Bergman spaces of the unit disk, recall the following consequence of Stanton’s formula [28, 29]:

$$\begin{aligned} \left\| h\right\| ^{2}_{A_{\delta }\left( \mathbb {D}\right) }\asymp \left| h(0)\right| ^2+\int _{\mathbb {D}}\left| h'(z)\right| ^2 \left( 1-\left| z\right| ^2\right) ^{\delta +1}dV(z),\ \text { for }h \text { holomorphic on }\mathbb {D}. \end{aligned}$$

Via the mapping $\tau _{1/2}$, we obtain that, for any f holomorphic on $\mathbb {C}_{1/2}$,

$$\begin{aligned} \left\| f\right\| ^{2}_{A_{i,\delta }\left( \mathbb {C}_{1/2}\right) }\asymp \left| f(\frac{3}{2})\right| ^2+\int _{\mathbb {C}_{1/2}}\left| f'(s)\right| ^2 \frac{\left( \sigma -\frac{1}{2}\right) ^{\delta +1}}{\left| s+\frac{1}{2}\right| ^{2\delta +2}}dV(s). \end{aligned}$$

(3.1)

4 Boundedness of $T_g$

In this section, we characterize functions in ${{\mathcal {X}}_{w}}$, and prove Theorem 1.

4.1 Carleson measure characterization

The boundedness of $T_g$ on ${{\mathcal {H}}^{2}_{w}}$ can be described in terms of Carleson measures. This generalizes the setting of the Hardy space ${{\mathcal {H}}}^2$ [13].

Recall that ${{\mathcal {H}}^{2}_{w}}$ is associated to the probability measure $\mu _{w}$ on the polydisk $\mathbb {D}^{\infty }$.

Proposition 3

$T_g$ is bounded on ${{\mathcal {H}}^{2}_{w}}$ if and only if there exists a constant $C=C(g)$ such that

$$\begin{aligned} \left\| T_g f\right\| ^2_{{{\mathcal {H}}^{2}_{w}}}&\asymp&\int _{\mathbb {D}^{\infty }}\int _{\mathbb {R}}\int ^{+\infty }_{0}\left| f_{\chi }(\sigma +it)\right| ^2\left| g'_{\chi }(\sigma +it)\right| ^2\frac{\sigma d\sigma dt}{1+t^2}d\mu _w(\chi )\nonumber \\\le & {} C^2 \left\| f\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}, \end{aligned}$$

(4.1)

or, equivalently

$$\begin{aligned} \int _{\mathbb {D}^{\infty }}\int ^{+\infty }_{0} \left| f_{\chi }(\sigma )\right| ^2\left| g'_{\chi }(\sigma )\right| ^2 \sigma d\sigma d\mu _w(\chi )\le C^2 \left\| f\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}. \end{aligned}$$

(4.2)

The smallest constant C satisfying (4.1) is such that $C\asymp \left\| T_g\right\| _{{{\mathcal {L}}}({{\mathcal {H}}^{2}_{w}})}$.

Proof

Applying the Littlewood–Paley formula (Proposition  1) to the measure $d\lambda (t)=\pi ^{-1}(1+t^2)^{-1}dt$ and the function $T_g f$, we get (4.1).

The rotation invariance of the measure $d\mu _w(\chi )$ gives (4.2).

$\square$

4.2 Proof of Theorem 1 (a): ${{\mathcal {B}}}g$ depends on a finite number of variables

For $1\le q$ and $d\ge 1,$ recall that $f\in {{\mathcal {H}}}^q_{d,w}$ if and only if f is in ${{\mathcal {H}}}^q_{w}$ and ${{\mathcal {B}}}f$ is a function of $z_1,\ldots ,z_d$.

When needed, we shall identify $z=(z_1,\ldots ,z_d)\in \mathbb {D}^d$ with $(z,0)\in \mathbb {D}^d\times \left\{ 0\right\}$.

If $g(s)= \sum ^{+\infty }_{n=2}b_n n^{-s}$ is in ${{\mathcal {H}}}^2_{d,w}$, we observe that for $z\in \mathbb {D}^d,$

$$\begin{aligned} {{\mathcal {B}}}g'(z) =\sum ^{d}_{j=1}\log p_j \sum _{\alpha \in \mathbb {N}^d}{\tilde{b}}_{\alpha }\alpha _j z^{\alpha } =R {{\mathcal {B}}}g(z), \end{aligned}$$

where R is the operator

$$\begin{aligned} RG(z_1,\ldots ,z_d)=\sum ^{d}_{j=1}(\log p_j) z_j \partial _j G(z_1,\ldots ,z_d). \end{aligned}$$

We define the set

$$\begin{aligned} \Delta _{\epsilon }:=\left\{ z=(z_1,\ldots ,z_d)\in \mathbb {D}^d,\ \forall j,\ \left| z_j\right| < p^{-\epsilon }_{j}\right\} ,\ \text { for }\epsilon >0. \end{aligned}$$

Take $x>0$, $t\in \mathbb {R}$, and $z\in \mathbb {D}^d$. By construction, $z\in \overline{\Delta _{\sigma (z)}}$ and $\sigma ({{\mathfrak {p}}^{-\mathbf{x} }}.z)\ge \sigma (z)+x\frac{\log p_1}{\log p_d}$.

For $g\in {{\mathcal {D}}}_{d}$, we write $g_z(x)=g_{(z,0)}(x)={{\mathcal {B}}}g_x(z).$ Since g is in $\text {Bloch}(\mathbb {C}_0)$, we apply (1.6) to $g'_x$, and get

$$\begin{aligned} \left| g'_z(x+it)\right|= & {} \left| {{\mathcal {B}}}g'_x({{\mathcal {T}}}_t z)\right| \le \sup _{\zeta \in \overline{\Delta _{\sigma ({{\mathfrak {p}}^{-\mathbf{x} }}.z)}}} \left| {{\mathcal {B}}}g'(\zeta )\right| \nonumber \\= & {} \sup _{s\in \overline{\mathbb {C}_{\sigma ({{\mathfrak {p}}^{-\mathbf{x} }}.z)}}} \left| g'(s)\right| \le \frac{\log p_d}{\log p_1}\frac{\left\| g\right\| _{\text {Bloch}(\mathbb {C}_0)}}{x+\sigma (z)}, \end{aligned}$$

(4.3)

Proof of Theorem 1(a)

Let $f(s)=\sum _{n\ge 1}a_n n^{-s}$ be in ${{\mathcal {H}}}^2_{w}$, and, for $\chi =(z,z')\in \mathbb {D}^d\times \mathbb {D}^{\infty },$

$$\begin{aligned} {{\mathcal {B}}}f(\chi ) =\sum _{(\alpha ,\alpha ')\in \mathbb {N}^d\times \mathbb {N}^{\infty }_{\text {0,fin}}}c_{\alpha ,\alpha '}z^{\alpha }z'^{\alpha '}=\sum _{\alpha \in \mathbb {N}^d}c'_{\alpha }(z')z^{\alpha }, \text { where }c'_{\alpha }(z')=\sum _{\alpha '\in \mathbb {N}^{\infty }_{\text {0,fin}}}c_{\alpha ,\alpha '}z'^{\alpha '}. \end{aligned}$$

In view of Proposition 3, we aim to estimate $\left\| T_g f\right\| ^{2}_{{{\mathcal {H}}}^2_{w}}\asymp {{\mathcal {I}}}_1+{{\mathcal {I}}}_2,$ where

$$\begin{aligned} {{\mathcal {I}}}_1&:= \int _{\mathbb {D}^{\infty }}\int ^{1}_{0} \left| f_{\chi }(x)\right| ^2\left| g'_{\chi }(x)\right| ^2 x dx d\mu _w(\chi ),\\ \text { and } {{\mathcal {I}}}_2&:= \int _{\mathbb {D}^{\infty }}\int ^{+\infty }_{1} \left| f_{\chi }(x)\right| ^2\left| g'_{\chi }(x)\right| ^2 x dx d\mu _w(\chi ). \end{aligned}$$

By (4.3), the rotation invariance and Fubini’s Theorem, we have

$$\begin{aligned} {{\mathcal {I}}}_1&\lesssim \left\| g\right\| _{\text {Bloch}(\mathbb {C}_0)}^2 \int ^{1}_{0}x\int _{\mathbb {D}^{\infty }}\int _{\mathbb {D}^d}\frac{1}{\left[ x+\sigma (z)\right] ^2}\\&\quad \left| \sum _{\alpha \in \mathbb {N}^d}c'_{\alpha }({{\mathfrak {p}}'^{-\mathbf{x} }}. z')\left( z_1 p^{-x}_{1}\right) ^{\alpha _1}\cdots \left( z_d p^{-x}_{d}\right) ^{\alpha _d}\right| ^2 d\mu _w(z,z')dx\\&\lesssim \left\| g\right\| _{\text {Bloch}(\mathbb {C}_0)}^2 \int _{\mathbb {D}^{\infty }} \int ^{1}_{0}x\sum _{\alpha \in \mathbb {N}^d} \left| c'_{\alpha }({{\mathfrak {p}}'^{-\mathbf{x} }}. z')\right| ^2 I_\alpha (x) dx d\mu _w(z'), \end{aligned}$$

where

$$\begin{aligned} I_\alpha (x) :=\int _{\mathbb {D}^d}\frac{1}{\left[ x+\sigma (z)\right] ^2}\left| z_1 p^{-x}_{1}\right| ^{2\alpha _1}\cdots \left| z_d p^{-x}_{d}\right| ^{2\alpha _d}d\mu _w(z). \end{aligned}$$

Using the rotation invariance again as well as the fact that $p_j\ge 1$, and setting ${{\mathcal {J}}}_{\alpha }:=\int ^{1}_{0}x I_\alpha (x) dx$, we get

$$\begin{aligned} {{\mathcal {I}}}_1&\lesssim \left\| g\right\| _{\text {Bloch}(\mathbb {C}_0)}^2 \sum _{\alpha \in \mathbb {N}^d} \int ^{1}_{0}x I_\alpha (x) \left( \int _{\mathbb {D}^{\infty }}\left| \sum _{\alpha '} c_{\alpha ,\alpha '}({{\mathfrak {p}}'^{-\mathbf{x} }}. z')^{\alpha '}\right| ^2 d\mu _w(z') \right) dx\\&\lesssim \left\| g\right\| _{\text {Bloch}(\mathbb {C}_0)}^2 \sum _{\alpha ,\alpha '}\left| c_{\alpha ,\alpha '}\right| ^2 {{\mathcal {J}}}_{\alpha }\left( \int _{\mathbb {D}^{\infty }}\left| z'^{\alpha '}\right| ^2 d\mu _w(z') \right) \\&\lesssim \left\| g\right\| _{\text {Bloch}(\mathbb {C}_0)}^2 \sum _{\alpha ,\alpha '}\frac{\left| c_{\alpha ,\alpha '}\right| ^2 {{\mathcal {J}}}_{\alpha }}{w\left( p^{ \alpha _{d+1}}_{d+1}\right) \cdots w\left( p^{ \alpha _{r}}_{r}\right) } . \end{aligned}$$

For the moment, we admit that ${{\mathcal {J}}}_{\alpha }\le C(d,w)\left[ \prod _{j=1}^{d}{w(p^{\alpha _j}_{j})}\right] ^{-1},$ which will be proved in Lemma 7. Hence,

$$\begin{aligned} {{\mathcal {I}}}_1\lesssim \left\| g\right\| _{\text {Bloch}(\mathbb {C}_0)}^2 \sum _{\alpha ,\alpha '} \frac{\left| c_{\alpha ,\alpha '}\right| ^2}{w(p^{(\alpha ,\alpha ')})} \lesssim \left\| g\right\| _{\text {Bloch}(\mathbb {C}_0)}^2 \left\| f\right\| ^{2}_{{{\mathcal {H}}}^{2}_{w}}. \end{aligned}$$

Combining Lemma 3 with the following observation,

$$\begin{aligned} \int _{\mathbb {D}^{\infty }}\left| f_{\chi }(x)\right| ^2 d\mu _w(\chi )= & {} \int _{\mathbb {D}^{\infty }}\left| \sum _{n=p^{\alpha }}a_n n^{-x}\chi ^{\alpha }\right| ^2 d\mu _w(\chi ) \\= & {} \sum _{n\ge 1}\frac{\left| a_n\right| ^2 n^{-2x}}{w_n}\le \left\| f\right\| ^{2}_{{{\mathcal {H}}}^{2}_{w}}, \end{aligned}$$

we estimate ${{\mathcal {I}}}_2$,

$$\begin{aligned} {{\mathcal {I}}}_2 \lesssim \int ^{+\infty }_{1}x \int _{\mathbb {D}^{\infty }}\left\| g\right\| ^{2}_{\text {Bloch}(\mathbb {C}_0)}4^{-x} \left| f_{\chi }(x)\right| ^2 d\mu _w(\chi )dx \lesssim \left\| g\right\| ^{2}|_{\text {Bloch}(\mathbb {C}_0)}\left\| f\right\| ^{2}_{{{\mathcal {H}}}^{2}_{w}}. \end{aligned}$$

$\square$

Recall that

$$\begin{aligned} I_\alpha (x)=\int _{\mathbb {D}^d}\frac{1}{\left[ x+\sigma (z)\right] ^2}\left| z_1 p^{-x}_{1}\right| ^{2\alpha _1}\cdots \left| z_d p^{-x}_{d}\right| ^{2\alpha _d}d\mu _w(z),\ \alpha \in \mathbb {N}^d,\ 0<x<1. \end{aligned}$$

Lemma 7

There exists a constant $C=C(w,d)$, such that

$$\begin{aligned} {{\mathcal {J}}}_{\alpha }:= \int ^{1}_{0}xI_\alpha (x)dx\le C \prod ^{d}_{j=1} \frac{1}{w\left( p^{\alpha _j}_{j}\right) }. \end{aligned}$$

The proof of Lemma 7 relies on technical computations (Lemma 8).

Lemma 8

For $0<T<1$, and a real number $p\ge 2$, set $L:=-\frac{\log T}{2\log p}$ and $K=\min (1,L)$. There exists a constant $C=C(p,w)>0$, such that

$$\begin{aligned} J(p, T):= & {} \left( \log T\right) ^{-2}\int ^{K}_{0}xM\left( T p^{2x}\right) dx\\\lesssim & {} C{\left\{ \begin{array}{ll}M \left( T\right) &{}\text { if }\beta \ge 1\text { or }(\beta< 1, p^{-2}< T<1),\\ M\left( Tp^2\right) &{} \text { if }\beta< 1, 0<T\le p^{-2}. \end{array}\right. } \end{aligned}$$

Proof

When $p^{-2}< T<1$, the change of variables $u=T p^{2x}$ gives

$$\begin{aligned} J(p,T)&=\left( \log T\right) ^{-2}\frac{1}{(2\log p)^2} \int ^{1}_{T}\log \frac{u}{T}M(u)\frac{du}{u}. \end{aligned}$$

Since $\log \frac{u}{T}\le \log \frac{1}{T}$ and $1\le \frac{1}{u}\le \frac{1}{T}<p^{2}$,

$$\begin{aligned} J(p,T)&\le \left( \log T\right) ^{-2}\left( \frac{1}{2\log p}\right) ^2 \int ^{1}_{T}\log \frac{1}{T}M(u)\frac{1}{u}du\lesssim M(T). \end{aligned}$$

Next suppose that $0<T\le p^{-2}$. Since $(\log T)^2\ge 4(\log p)^2$, we notice that

$$\begin{aligned} J(p,T)\lesssim \int ^{1}_{0}xM(T p^{2x})dx \lesssim {\left\{ \begin{array}{ll} \int ^{1}_{0}M(T )dx\text { if }\beta \ge 1 ,\\ \int ^{1}_{0}M(T p^{2})dx \text { if }\beta < 1\end{array}\right. }. \end{aligned}$$

$\square$

Proof of Lemma 7

Resorting to polar coordinates, and using changes of variables, we have

$$\begin{aligned} {{\mathcal {J}}}_{\alpha }&\le \int _{Q}\frac{xt^{\alpha }}{\left[ x+\sigma \left( p^{x}_{1}\sqrt{t_1},\ldots ,p^{x}_{1}\sqrt{t_d}\right) \right] ^2} \left( \prod ^{d}_{k=1} M\left( p^{2x}_{k}t_{k}\right) p^{2x}_{k}\right) dxdt_1\cdots dt_d , \end{aligned}$$

where $Q=\left\{ (x,t)\in (0,1)\times (0,1)^d,\ \forall k=1..d, 0< t_k<p^{-2x}_{k} \right\}$.

For $t=(t_1,\ldots ,t_d)\in (0,1)^d$, set

$$\begin{aligned} l_k(t)&:=-\frac{\log t_k}{2\log p_k},\ K_k:=\min (1,l_k),\ 1\le k\le d,\\ l(t)&:=\min _{1\le k\le d}l_k(t),\ K:=\min (1,l). \end{aligned}$$

We observe that $Q =\left\{ (x,t)\in (0,1)\times (0,1)^d,\ 0<x<K(t)\right\}$. Now, for $1\le k\le d,$ we set $Q_k:=\left\{ (x,t),\ t\in (0,1)^d,\ l(t)=l_k(t),\ 0<x<K_k(t)\right\} .$

Let (x, t) be in $Q_k$. We have

$$\begin{aligned} 0<t_l\le T_{k,l}:=t^{\frac{\log p_l}{\log p_k}}_{k}<1,\ \text { for }1\le l\le d. \end{aligned}$$

(4.4)

In addition, since $0< x< l_k(t)$, (4.4) implies $p^{x}_{l}\sqrt{t_l}<p^{l_k(t)}_{l}\sqrt{t_l} \le 1,$ and we see that $\frac{1}{\sqrt{t_l} p^{x}_{l}} \ge p^{l_k(t)-x}_{l}\ge p^{l_k(t)-x}_{1}.$ Thus

$$\begin{aligned} (\log p_d ) \sigma \left( p^{x}_{1}\sqrt{t_1},\ldots ,p^{x}_{d}\sqrt{t_d}\right)&=\log \min _{1\le l\le d}\left( \frac{1}{\sqrt{t_l}p^{x}_{l}}\right) \ge \log p_1 \left( l_k(t)-x \right) , \end{aligned}$$

and $x+\sigma \left( p^{x}_{1}\sqrt{t_1},\ldots ,p^{x}_{1}\sqrt{t_d}\right) \gtrsim -\log t_k.$

Set $d\widehat{t_k}=dt_1\cdots dt_{k-1}dt_{k+1}\cdots dt_d,$ and

$$\begin{aligned} {\tilde{Q}}_k:=\left\{ (x,t),\ 0<t_k<1,\ 0<t_l<T_{k,l} \text { for }l\ne k,\ 0<x<K_k(t)\right\} . \end{aligned}$$

It follows that ${{\mathcal {J}}}_{\alpha } \lesssim \sum ^{d}_{k=1}{{\mathcal {J}}}_{\alpha ,k}$, where

$$\begin{aligned} {{\mathcal {J}}}_{\alpha ,k}&=\int _{{\tilde{Q}}_k}\frac{x t^{\alpha }}{\left[ x+\sigma \left( p^{x}_{1}\sqrt{t_1},\ldots ,p^{x}_{1}\sqrt{t_d}\right) \right] ^2}\left( \prod ^{d}_{l=1} M \left( p^{2x}_{l}t_{l}\right) \right) dx dt. \end{aligned}$$

We will obtain the Lemma by showing that

$$\begin{aligned} {{\mathcal {J}}}_{\alpha ,k}\lesssim \prod ^{d}_{l=1}\left[ {w\left( p^{\alpha _l}_{l}\right) }\right] ^{-1}. \end{aligned}$$

(4.5)

When $\beta \ge 1$, we use that, for $(x,t)\in {\tilde{Q}}_k$, and $l\ne k$, $M\left( p^{2x}_{l}t_{l}\right) \le M\left( t_{l}\right)$, altogether with Lemma 8. We derive (4.5) from

$$\begin{aligned} {{\mathcal {J}}}_{\alpha ,k}&\lesssim \int _{0<t_k<1}\left( \int _{\prod _{j\ne k}(0,T_{k,j})}t^{\alpha }\int ^{K_k(t)}_{0}x\left( \log t_k\right) ^{-2}M\left( p^{2x}_{k}t_{k}\right) dx \prod _{l\ne k}M (t_l)d\widehat{t_k} \right) dt_k\\&\lesssim \int _{0<t_k<1}t^{\alpha _k}_{k}M\left( t_{k}\right) \left( \prod _{j\ne k}\int ^{T_{k,j}}_{0}t^{\alpha _j}_{j}M\left( t_{j}\right) dt_j\right) dt_k \lesssim \prod ^{d}_{j=1}\int ^{1}_{0}t^{\alpha _j}_{j}M\left( t_{j}\right) dt_j. \end{aligned}$$

Next, suppose $0<\beta <1$. If $(x,t)\in {\tilde{Q}}_k$, notice that, for $l\ne k$, $t_l p^{2x}_{l} \le t_l p^{2 l_k(t)}_{l}\le 1$; this shows that $M\left( p^{2x}_{l}t_{l}\right) \le M\left( p^{2 l_k(t) }_{l}t_{l}\right) .$ Hence, we see that ${{\mathcal {J}}}_{\alpha ,k} \lesssim J_1+J_2$, where, by Lemma 8 and the relation $p^{2 l_k(t) }_{l}=T^{-1}_{k,l}$,

$$\begin{aligned} J_1&\lesssim \int _{0<t_k< p^{-2}_{k}}t^{\alpha _k}_{k}M(p^{2 }_{k}t_k)\left( \prod _{j\ne k}\int ^{T_{k,j}}_{0}t^{\alpha _j}_{j}M\left( t_{j}T_{k,j}^{-1}\right) dt_j \right) dt_k,\\ J_2&\lesssim \int _{p^{-2}_{k}<t_k <1}t^{\alpha _k}_{k}M(t_k)\left( \prod _{j\ne k}\int ^{T_{k,j}}_{0}t^{\alpha _j}_{j}M\left( t_{j}T_{k,j}^{-1}\right) dt_j \right) dt_k. \end{aligned}$$

A change of variables provides the desired estimate.

$\square$

4.3 Proof of Theorem 1(b) and (c)

If $f(s)= \sum ^{+\infty }_{n=1}a_n n^{-s}$ and $g(s)= \sum ^{+\infty }_{n=1}b_n n^{-s}$, we have

$$\begin{aligned} T_g f(s)=\sum ^{\infty }_{n=2}\frac{1}{\log n}\left( \sum _{k|n,k<n}a_k b_{n/k}\right) n^{-s}. \end{aligned}$$

As in the case of ${{\mathcal {H}}}^2$, the operator

$$\begin{aligned} a_1+ \sum ^{\infty }_{n=2}a_n n^{-s}\mapsto a_1+\sum ^{\infty }_{n=2}a_n (\log n)^{-1}n^{-s} \end{aligned}$$

is compact on ${{\mathcal {H}}}_w$. Thus, set $b_1=1$, and our study will be unchanged if we replace $T_g$ by

$$\begin{aligned} \tilde{T}_g f(s)=\sum ^{\infty }_{n=2}\frac{1}{\log n}\left( \sum _{k|n}a_k b_{n/k}\right) n^{-s}. \end{aligned}$$

Lemma 9

If $T_g$ is bounded on ${{\mathcal {H}}}^2$, then g is in ${{\mathcal {X}}_{w}}$, and the operator norms satisfy

$$\begin{aligned} \left\| T_g\right\| _{{{\mathcal {L}}}({{\mathcal {H}}}^2_w)}\le \left\| T_g\right\| _{{{\mathcal {L}}}({{\mathcal {H}}}^2)}. \end{aligned}$$

Proof

If $f(s)= \sum ^{+\infty }_{n=1}a_n n^{-s}$ is in ${{\mathcal {H}}}^2_w$, the function ${\tilde{f}}(s)= \sum ^{+\infty }_{n=1}a_n w^{-1/2}_{n} n^{-s}$ is in ${{\mathcal {H}}}^2$ and $\left\| f\right\| _{{{\mathcal {H}}^{2}_{w}}}=\left\| {\tilde{f}}\right\| _{{{\mathcal {H}}}^2}$. Since $w_k\le w_{kl}$ for any integers k, l, the Lemma is proven by the inequality

$$\begin{aligned} \left\| T_g f\right\| ^{2}_{{{\mathcal {H}}}^2_w}\le \sum ^{\infty }_{n=2}\left( \log n\right) ^{-2}\left| \sum _{k|n,k<n}w^{-1/2}_{k}a_k b_{n/k}\right| ^2=\left\| T_g {\tilde{f}}\right\| ^{2}_{{{\mathcal {H}}}^2}. \end{aligned}$$

$\square$

We will also use the sufficient condition proved in Theorem 2.3 in [13], stating that if g is in $BMOA(\mathbb {C}_0)\cap {{\mathcal {D}}}$, then $T_g$ is bounded on ${{\mathcal {H}}}^2$, with

$$\begin{aligned} \left\| T_g\right\| _{{{\mathcal {H}}}^2}\lesssim \left\| g\right\| _{BMOA(\mathbb {C}_0)}. \end{aligned}$$

(4.6)

Proof of Theorem 1(b) and (c)

If g is in $BMOA(\mathbb {C}_0)$, $T_g$ is bounded on ${{\mathcal {H}}}^2$, and (b) is a consequence of (4.6) and Lemma 9.

To prove (c), we use that $(T_g f)' =fg'$, and that ${{\mathcal {H}}^{2}_{w}}$ is embedded in $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$, with $\delta =\delta (w)>0.$ We set

$$\begin{aligned} d\nu _g(s)=\left| g'(s)\right| ^2 \frac{\left( \sigma -\frac{1}{2}\right) ^{\delta +1}}{\left| s+\frac{1}{2}\right| ^{2\left( \delta +1\right) }}dV(s). \end{aligned}$$

Now formula (3.1), the boundedness of $T_g$ on ${{\mathcal {H}}^{2}_{w}}$ and Lemma  1 induce that

$$\begin{aligned} \int _{\mathbb {C}_{1/2}}\left| f(s)\right| ^2 d\nu _g(s)\lesssim \left\| T_g f\right\| ^{2}_{A_{i,\delta }\left( \mathbb {C}_{1/2}\right) } \le c\left( w\right) \left\| T_gf\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}\le c\left( w\right) \left\| T_g\right\| ^{2}_{{{\mathcal {L}}}\left( {{\mathcal {H}}^{2}_{w}}\right) }\left\| f\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}} , \end{aligned}$$

Thus, $\nu _g$ is a Carleson measure for ${{\mathcal {H}}^{2}_{w}}$ and $\left\| \nu _g\right\| _{CM\left( {{\mathcal {H}}^{2}_{w}}\right) }\lesssim \left\| T_g\right\| ^{2}_{{{\mathcal {L}}}\left( {{\mathcal {H}}^{2}_{w}}\right) }$. By Lemma 6, $\nu _g$ is also a Carleson measure for $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$ and

$$\begin{aligned} \left\| \nu _g\right\| _{CM\left( A_{i,\delta }\left( \mathbb {C}_{1/2}\right) \right) } \lesssim \left\| T_g\right\| ^{2}_{{{\mathcal {L}}}\left( {{\mathcal {H}}^{2}_{w}}\right) }. \end{aligned}$$

We conclude by the characterization of the Bloch space given in Lemma 4.

$\square$

We get a result which is in agreement with the situation for Hardy spaces [15], Bergman spaces [2] or the Hardy space of Dirichlet series ${{\mathcal {H}}}^2$ [13], with the same proof.

Corollary 1

If g is in ${{\mathcal {X}}_{w}}$, then g is in $\cap _{0<p<\infty }{{\mathcal {H}}}^p_{w}$, and there exists $c>0$, such that the function $e^{c\left| {{\mathcal {B}}}g\right| }$ is integrable on $\mathbb {D}^{\infty }$, with respect to $d \mu _{w}$.

5 Compactness

We now present a little oh version of Theorem 1.

If the symbol is a vector of the standard orthonormal basis of ${{\mathcal {H}}^{2}_{w}}$, that is

$$\begin{aligned} g(s)= e_{w,n}(s):= w^{1/2}_{n}n^{-s}, \end{aligned}$$

the operator $T^{*}_{g}T_g$ is diagonal, and its eigenvalues

$$\begin{aligned} \lambda _{n,k}^{2}=\frac{w_n w_k}{w_{nk}}\left( \frac{\log n}{\log n+\log k}\right) ^2 \end{aligned}$$

tend to 0 as $k\rightarrow +\infty$. Thus $T_g$ is compact. It follows that every Dirichlet polynomial generates a compact Volterra operator on ${{\mathcal {H}}^{2}_{w}}$.

5.1 Case when ${{\mathcal {B}}}g$ depends on a finite number of variables

We approximate a symbol g which is in $\text {Bloch}_0(\mathbb {C}_{0})\cap {{\mathcal {D}}}_{d}$ by a Dirichlet polynomial P in the $\text {Bloch}(\mathbb {C}_{0})$-norm. From Theorem 1(a), $T_g$ is approximated in the operator norm by the compact operator $T_P$.

Theorem 2

If g is in $\text {Bloch}_0(\mathbb {C}_{0})\cap {{\mathcal {D}}}_{d}$, then $T_g$ is compact on ${{\mathcal {H}}^{2}_{w}}$.

5.2 Sufficient/necessary conditions for compactness

In general, if the symbol $g(s)=\sum _{n\ge 2}b_n n^{-s}$ satisfies an inequality of the form $\left\| T_g\right\| ^{2}_{{{\mathcal {L}}}({{\mathcal {H}}^{2}_{w}})}\le \sum _{n\ge 2} \left| b_n\right| ^2 W(n)<\infty$, we approximate $T_g$ in the operator norm by the compact operator $T_{S_N g}$. Therefore, $T_g$ is compact (see [13]).

The little oh version of Theorem 1 is related to the properties of $VMOA(\mathbb {C}_0)\cap {{\mathcal {D}}}$, and with the concept of vanishing Carleson measures.

Theorem 3

Let g be in ${{\mathcal {D}}}$.

• (1) If g is in $VMOA(\mathbb {C}_0)\cap {{\mathcal {D}}}$, then $T_g$ is compact on ${{\mathcal {H}}^{2}_{w}}$.

• (2) If $T_g$ is compact on ${{\mathcal {H}}^{2}_{w}}$, then g is in $\text {Bloch}_0(\mathbb {C}_{1/2})$.

Proof

In order to prove (1), we use that $VMOA(\mathbb {C}_0)\cap {{\mathcal {D}}}$ is the closure of Dirichlet polynomials in $BMOA(\mathbb {C}_0)$ (see [13]), and that, from Theorem  1, we have $\left\| T_g\right\| _{{{\mathcal {L}}}({{\mathcal {H}}^{2}_{w}})}\lesssim \left\| g\right\| _{BMOA(\mathbb {C}_0)}$.

Recall that ${{\mathcal {H}}^{2}_{w}}$ is embedded in $A_{i,\delta }(\mathbb {C}_{1/2})$, $\delta =\delta (w)$ being defined in (2.5). Assume that $T_g$ is compact on ${{\mathcal {H}}^{2}_{w}}$, and consider the measure

$$\begin{aligned} d\nu _g(s)=\left| g'(s)\right| ^2 \frac{\left( \sigma -\frac{1}{2}\right) ^{\delta +1}}{\left| s+\frac{1}{2}\right| ^{2(\delta +1)}}dV(s). \end{aligned}$$

Let $(f_k)_k$ be a weakly compact sequence in ${{\mathcal {H}}^{2}_{w}}$. Formula (3.1), and Lemma  1 imply that

$$\begin{aligned} \int _{\mathbb {C}_{1/2}}\left| f_k(s)\right| ^2 d\nu _g(s)\asymp \left\| T_g f_k\right\| ^{2}_{A_{i, \delta }(\mathbb {C}_{1/2})}\lesssim \left\| T_g f_k\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}. \end{aligned}$$

By the compactness of $T_g$, $\nu _g$ is a vanishing Carleson measure for $A_{i,\delta }(\mathbb {C}_{1/2})$, with

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _{\mathbb {C}_{1/2}}\left| f_k(s)\right| ^2 d\nu _g(s)=0. \end{aligned}$$

Now, g is in $\text {Bloch}_0(\mathbb {C}_{1/2})$, by the characterization of vanishing Carleson measures (Lemma 5).

$\square$

6 Membership in Schatten classes

Let g be a non constant symbol. As in the case of ${{\mathcal {H}}}^2$, the Volterra operator $T_g$ on ${{\mathcal {H}}^{2}_{w}}$ does not belong to any Schatten class.

Theorem 4

If the Dirichlet series $g(s)=\sum _{n\ge 2}b_n n^{-s}$ is not 0, then $T_g:{{\mathcal {H}}^{2}_{w}}\rightarrow {{\mathcal {H}}^{2}_{w}}$ is not in the Schatten class ${{\mathcal {S}}}_p$, for any $0<p<\infty$.

Proof

Recall that $(e_{w,n})_n$ is an orthonormal basis of ${{\mathcal {H}}^{2}_{w}}$. We follow the reasoning of Theorem 7.2 [13]. Using that $w_{Nn}\lesssim w_{N}w_{n}$, we see that, for $N\ge n$,

$$\begin{aligned} \left\| T_g e_{w,n}\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}&=\sum ^{+\infty }_{k=2}\frac{\left| b_k\right| ^2 (\log k)^2}{(\log (kn))^2}\frac{w_n}{w_{kn}} \ge \frac{\left| b_N\right| ^2 (\log N)^2}{(\log (Nn))^2}\frac{w_n}{w_{Nn}}\gtrsim \frac{\left| b_N\right| ^2 (\log N)^2}{(2\log n)^2}\frac{1}{w_{N}}. \end{aligned}$$

For $p\ge 2$, we obtain

$$\begin{aligned} \left\| T_g \right\| ^{p}_{{{\mathcal {S}}}_p}\ge \sum ^{+\infty }_{n=N}\left\| T_g e_{w,n}\right\| ^{p}_{{{\mathcal {H}}^{2}_{w}}}=+\infty . \end{aligned}$$

Therefore $T_g$ is not in ${{\mathcal {S}}}_p$ for $p\ge 2$, neither for $0<p<\infty$. $\square$

7 Examples

In this section, we study the boundedness of $T_g$ on ${{\mathcal {H}}^{2}_{w}}$ , for specific symbols g. We consider fractional primitives of translates of the weighted Zeta function $Z_w$ and homogeneous symbols, which are the counterparts of the symbols presented in [13] in the ${{\mathcal {H}}}^2$ setting. The techniques of proof, as well as the results are similar to theirs, and we omit the details.

7.1 Fractional primitives of translates of $Z_w$

Proposition 4

With the notation of (2.5), take $1/2\le a<1$, $2\gamma >\delta (w)-1$. If

$$\begin{aligned} g(s)=\sum ^{\infty }_{n=2}w_n\frac{n^{-a}}{\left( \log n\right) ^{\gamma +1}} n^{-s}, \end{aligned}$$

then $T_g$ is unbounded on ${{\mathcal {H}}^{2}_{w}}$.

Proof

Abel summation and the Chebyshev estimate induce that g is in ${{\mathcal {H}}^{2}_{w}}$. If $f (s)=\sum ^{\infty }_{n=1}a_n n^{-s}$, and $g(s)=\sum ^{\infty }_{n=2}\frac{b_n}{\log n}n^{-s}$, we set $A_n=\sum _{k|n}a_{n/k}b_k$, so that

$$\begin{aligned} \left\| \tilde{T}_{g}f\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}=\sum ^{\infty }_{n=2}\frac{1}{(w_n\log n)^2}A^2_n. \end{aligned}$$

We adapt the test functions of [13], and take $f_J(s)=\prod ^{J}_{j=1}\left( 1+w^{1/2}_{2}p^{-s}_{j}\right)$, for $J\ge 1$. By construction, it satisfies $\left\| f_J\right\| _{{{\mathcal {H}}^{2}_{w}}}\asymp 2^{J/2}$. Now, for ${\mathcal {J}}$ a non-empty subset of $\left\{ 1,\ldots , J\right\}$, we set $n_{{\mathcal {J}}}=\prod _{j\in {\mathcal {J}}}p_j$, and observe that

$$\begin{aligned} A_{n_{{\mathcal {J}}}} =\sum _{1\le k\le \left| {\mathcal {J}}\right| ,\left\{ p_{j_1},\ldots , p_{j_k}\right\} \subset {\mathcal {J}} }w^{\frac{\left| {\mathcal {J}}\right| -k}{2}}_{2}\left[ \log \left( p_{j_1 }\cdots p_{j_k }\right) \right] ^{-\gamma }w^{k}_{2}\left( p_{j_1 }\cdots p_{j_k }\right) ^{-a}+w^{\frac{\left| {\mathcal {J}}\right| }{2}}_{2}. \end{aligned}$$

First assume that $\gamma \ge 0$. From the prime number Theorem, we obtain that

$$\begin{aligned} A_{n_{{\mathcal {J}}}}&\gtrsim w^{\frac{\left| {\mathcal {J}}\right| }{2}}_{2}\left[ J\log J\right] ^{-\gamma } \left[ 1+\sum _{1\le k\le \left| {\mathcal {J}}\right| ,\left\{ p_{j_1},\ldots , p_{j_k}\right\} \subset {\mathcal {J}} } w_2^{k/2}\left( p_{j_1 }\cdots p_{j_k }\right) ^{-a}\right] . \end{aligned}$$

Therefore, it follows again from the prime number Theorem that

$$\begin{aligned} \left\| \tilde{T}_{g}f_J\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}&\gtrsim \sum _{{\mathcal {J}}\subset \left\{ 1,\ldots , J\right\} ,\left| {\mathcal {J}}\right| \ge J/2 }\frac{1}{\left( \log n_{{\mathcal {J}}}\right) ^2}\left[ J\log J\right] ^{-2\gamma } \prod _{j\in {\mathcal {J}}}\left( 1+w_2^{1/2} p_j^{-a}\right) ^2\\&\gtrsim 2^{J-1} \left[ J\log J\right] ^{-2\gamma } \min _{\left| {\mathcal {J}}\right| \ge J/2 }\frac{1}{\left( \log n_{{\mathcal {J}}}\right) ^2}\prod _{j\in {\mathcal {J}}}\left( 1+ w_2^{1/2} p_j^{-a}\right) ^2\\&\gtrsim e^{cJ^{1-a}\left( \log J\right) ^{-a}}\left\| f_J\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}, \end{aligned}$$

for some constant $c>0$, and $T_g$ is unbounded. The case when $\gamma <0$ is similar. $\square$

7.2 Homogeneous symbols

An m-homogeneous Dirichlet series has the form

$$\begin{aligned} g(s)=\sum _{\Omega (n)=m}b_n n^{-s}. \end{aligned}$$

We extend Theorem 4.2 in [13] to the spaces ${{\mathcal {H}}^{2}_{w}}$.

Proposition 5

There exist weights $W_m(n)$ such that for $g(s)=\sum _{\Omega (n)=m} b_n n^{-s},$

$$\begin{aligned} \left\| T_g\right\| _{{{\mathcal {L}}}({{\mathcal {H}}^{2}_{w}})}\le \left( \sum _{\Omega (n)=m}\left| b_n\right| ^2 W_m(n)\right) ^{1/2}. \end{aligned}$$

(7.1)

Precisely, there exist absolute constants $C_m$ for which

$$\begin{aligned} W_m(n)={\left\{ \begin{array}{ll} C_1 &{}\text { for }m=1,\\ C_2 \frac{\log n}{\log _2 n} &{} \text { for }m=2,\\ C_m \frac{ n^{\frac{m-2}{m}}}{\left( \log n\right) ^{m-2}}&{} \text { for }m\ge 3. \end{array}\right. } \end{aligned}$$

Moreover, when $m=2$, $\log _2 n$ cannot be replaced in (7.1) by $\left( \log _2 n\right) ^{1+{\varepsilon }}$ for any ${\varepsilon }>0.$

Proof

If a linear symbol ($m=1$) $g(s)=\sum _{p\in \mathbb {P}}b_p p^{-s}$ belongs to ${{\mathcal {H}}}^2$, we observe that $\left\| g\right\| ^2_{{{\mathcal {H}}}^2}=2^{\beta } \left\| g \right\| ^{2}_{{{\mathcal {B}}}^{2}_{\beta }}=\left( \beta +1\right) \left\| g\right\| ^{2}_{{{\mathcal {A}}}^{2}_{\beta }}$ . Hence, it follows from Theorem 4.1 in [13] and Lemma 9 that $T_{g}$ is bounded on ${{\mathcal {H}}^{2}_{w}}$ and $\left\| T_{g}\right\| _{{{\mathcal {L}}}({{\mathcal {H}}^{2}_{w}})}\le \left\| T_g\right\| _{{{\mathcal {L}}}({{\mathcal {H}}}^2)}.$ One can choose $C_1=\max \left( \left( \beta +1\right) ^{-1}, 2^{-\beta }\right)$.

(7.1) is a consequence of Theorem 4.2 in [13] and Lemma 9. We now prove the sharpness of the factor $\log _2 n$. We assume that for some ${\varepsilon }>0$, every 2-homogeneous Dirichlet series g satisfies

$$\begin{aligned} \left\| T_{g}\right\| _{{{\mathcal {L}}}({{\mathcal {H}}^{2}_{w}})}\le C_2 \left( \sum _{\Omega (n)=m}\left| b_n\right| ^2 \frac{\log n}{\left( \log _2 n\right) ^{1+{\varepsilon }}}\right) ^{1/2}. \end{aligned}$$

(7.2)

For x a large real number, and $q\sim e^x$ a prime number, the symbol considered in [13] is

$$\begin{aligned} g_x(s)=\sum _{x/2<p\le x}\frac{\left( \log _2 (pq)\right) ^{1+{\varepsilon }/2}}{p}\left( pq\right) ^{-s}. \end{aligned}$$

We take as test functions

$$\begin{aligned} f_x(s)=\sum ^{+\infty }_{n=1}a_n n^{-s}=\prod _{x/2<p\le x}\left( 1+ w^{1/2}_{2} p^{-s}\right) . \end{aligned}$$

If $S_x$ denotes the set of square-free integers generated by the primes $x/2<p\le x$, we have $\left\| f_x\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}\asymp \left| S_x\right| =2^{N(x)}$, where $N(x):=\pi (x)-\pi (x/2)$. Now,

$$\begin{aligned} \frac{\left\| T_{g_x}f_x\right\| ^2_{{{\mathcal {H}}^{2}_{w}}}}{\left\| f_x\right\| ^2_{{{\mathcal {H}}^{2}_{w}}}}\gtrsim \frac{1}{\left| S_x\right| } \sum _{n\in S_x}w^{-1}_{nq}\left( \log (nq)\right) ^{-2}\left| \sum _{pq|nq}\log (pq)\frac{\left( \log _2 (pq)\right) ^{1+{\varepsilon }/2}}{p}a_{n/p}\right| ^2. \end{aligned}$$

If $n\in S_x$, and p|n, we have $a_{n/p}=w_2^{\frac{1}{2}\left[ \omega (n)-1\right] },\ w_n=w^{\omega (n)}_{2},$ and $w_{nq}=w_{n} w_{q}$. Thus,

$$\begin{aligned} \frac{\left\| T_{g_x}f_x\right\| ^2_{{{\mathcal {H}}^{2}_{w}}}}{\left\| f_x\right\| ^2_{{{\mathcal {H}}^{2}_{w}}}} \gtrsim \frac{1}{\left| S_x\right| }\frac{\left( \log x\right) ^{2+{\varepsilon }}}{x^2} \sum _{n\in S_x}\omega (n)^2. \end{aligned}$$

Now $\sum _{n\in S_x}\omega (n)^2=\sum ^{N(x)}_{k=1}\left( {\begin{array}{c}N(x)\\ k\end{array}}\right) k^2\asymp N(x)^2 2^{N(x)}$, and (7.2) does not hold, due to

$$\begin{aligned} \frac{\left\| T_{g_x}f_x\right\| _{{{\mathcal {H}}^{2}_{w}}}}{\left\| f_x\right\| _{{{\mathcal {H}}^{2}_{w}}}} \gtrsim \left( \log x\right) ^{{\varepsilon }}. \end{aligned}$$

$\square$

We will exhibit an homogeneous symbol g which is in ${{\mathcal {H}}^{2}_{w}}\cap \text {Bloch}_0(\mathbb {C}_{1/2})$, but not in ${{\mathcal {X}}_{w}}$. In fact, we observe that g is in every ${{\mathcal {H}}^{p}_{w}}$.

Lemma 10

If g is an m-homogeneous Dirichlet series in ${{\mathcal {H}}^{2}_{w}}$, then g is in $\cap _{0<p<\infty }{{\mathcal {H}}}^{p}_{w}$ and, for any $0<p<\infty$, there exists $c=c(m,p)$ such that

$$\begin{aligned} \left\| g\right\| _{{{\mathcal {H}}}_w^{p}} \le c\left\| g\right\| _{{{\mathcal {H}}^{2}_{w}}}. \end{aligned}$$

(7.3)

Proof

It is enough to consider the case $p\ge 2$. We first prove the inequality for $p=2^k$, k being a positive integer, by an induction argument.

Obviously, it holds for $k=1$.

Our proof is inspired of Lemma 8 in [27]. For any integer m, there exists a constant C(m), such that $\max \left( w_n, d(n)\right) \le C(m)$, whenever $\Omega (n)=m$.

If $f(s)=\sum _{n}a_n n^{-s}$ is m-homogeneous, then $f^2(s)=\sum _{n}b_n n^{-s}$ is 2m-homogeneous, and $\left| b_n\right| ^2\le d(n) \sum _{k|n}\left| a_k\right| ^2\left| a_{n/k}\right| ^2 .$ Since $w_n\ge \sqrt{w_k}\sqrt{w_{n/k}},$

$$\begin{aligned} \left\| f\right\| ^{4}_{{{\mathcal {H}}}^{4}_{w}}&=\left\| f^2\right\| ^{2}_{{{\mathcal {H}}}^{2}_{w}}\le \sum _{\Omega (n)=2m}d(n)w^{-1}_{n} \left( \sum _{k|n}\left| a_k\right| ^2\left| a_{n/k}\right| ^2\right) \\&\le C(2m)\sum _{\Omega (n)=2m}\left( \sum _{k|n}\frac{\left| a_k\right| ^2}{\sqrt{w_k}}\frac{\left| a_{n/k}\right| ^2}{\sqrt{w_{n/k}}} \right) \\&=C(2m)\left( \sum _{k}\frac{\left| a_k\right| ^2}{\sqrt{w_k}}\right) ^2 \le C(2m) C(m)\left\| f\right\| ^{4}_{{{\mathcal {H}}}^{2}_{w}}. \end{aligned}$$

Now, suppose that, for some k, an m-homogeneous Dirichlet series h satisfies

$$\begin{aligned} \left\| h\right\| ^{2^k}_{{{\mathcal {H}}}^{2^k}_{w}}\le K(m,k) \left\| h\right\| ^{2^k}_{{{\mathcal {H}}}^{2}_{w}}\text { for any }m. \end{aligned}$$

We obtain that

$$\begin{aligned} \left\| f\right\| ^{2^{k+1}}_{{{\mathcal {H}}}^{2^{k+1}}_{w}}&= \left\| f^2\right\| ^{2^k}_{{{\mathcal {H}}}^{2^k}_{w}}\le K(2m,k) \left\| f^2\right\| ^{2^k}_{{{\mathcal {H}}}^{2}_{w}}= K(2m,k) \left\| f\right\| ^{2^{k+1}}_{{{\mathcal {H}}}^{4}_{w}}\\&\le K(2m,k)\left[ C(2m) C(m)\left\| f\right\| ^{4}_{{{\mathcal {H}}}^{2}_{w}}\right] ^{2^{k-1}}. \end{aligned}$$

For general p, (7.3) is a consequence of Hölder’s inequality. $\square$

For our construction, we need two technical Lemmas.

Lemma 11

Assume that $0<\delta <1$ and $0<\eta$. For $j=1..3$, we set $h_j(s)=\sum _{p\ge 3}\alpha _{j,p} p^{-s}$, where

$$\begin{aligned} \alpha _{1,p}=\left( \log _2 p\right) ^{-\delta },\ \alpha _{2,p}=\log _2 p,\ \alpha _{3,p}={\log p}{\left( \log _2 p\right) ^{-\eta }}. \end{aligned}$$

For a real number $\sigma >1$, set $\sigma ':=\frac{1}{\sigma -1}$. Then we have

$$\begin{aligned} h_1(\sigma )\asymp \left( \log \sigma '\right) ^{1-\delta };\ h_2(\sigma )\asymp \log _2\left( \sigma '\right) ;\ h_3(\sigma )\asymp \sigma '\left( \log \sigma '\right) ^{-\eta },\ \text { as }\sigma \rightarrow 1^+. \end{aligned}$$

(7.4)

Proof

These asymptotics will follow from computations inspired by [4, 20]. Recall that

$$\begin{aligned} A_1(t):=\sum _{3\le p\le t}\frac{1}{p}= \log _2 t+O(1). \end{aligned}$$

(7.5)

Setting $f_1(t)=\frac{t^{-(\sigma -1)}}{\left( \log _2 t\right) ^{\delta }}$, we have

$$\begin{aligned} h_1(\sigma )&= \sum _{p\ge 3}\frac{p^{-(\sigma -1)}}{p\left( \log _2 p\right) ^{\delta }}=-\int ^{+\infty }_{ 3}A_1(t) f'_1(t)dt+O(1)\\&\asymp (\sigma -1)\int ^{+\infty }_{3}\left( \log _2 t\right) ^{1-\delta }t^{-\sigma }dt \\&=(\sigma -1)\left( \int ^{\sigma '}_{\log 3}+\int ^{+\infty }_{\sigma '}\right) \left( \log x\right) ^{1-\delta }e^{-(\sigma -1)x}dx. \end{aligned}$$

Using integration by parts (for the first integral), and a change of variable (for the second one), we obtain

$$\begin{aligned} h_1(\sigma )&\asymp (\sigma -1) \int ^{\sigma '}_{\log 3}\left( \log x\right) ^{1-\delta }dx+\int ^{+\infty }_{1}\left( \log y+\log \sigma '\right) ^{1-\delta }e^{-y} dy\\&\asymp (\sigma -1)\left[ x\left( \log x\right) ^{1-\delta }\right] ^{x=\sigma '}_{x=\log 3}+\int ^{+\infty }_{1}\left[ \left( \log y\right) ^{1-\delta }+\left( \log \sigma '\right) ^{1-\delta }\right] e^{-y} dy \\&\asymp \left( \log \sigma '\right) ^{1-\delta }. \end{aligned}$$

The functions $h_2$ and $h_3$ are handled similarly. For $x\ge 3$, summation by parts and (7.5) induce that,

$$\begin{aligned} A_2(x):=\sum _{3\le p\le x}\frac{1}{p\log _2 p}=\frac{A_1(x)}{\log _2 x}+\int ^{x}_{3}\frac{A_1(t)}{t \log t (\log _2 t)^2}dt+O(1)\asymp \log _3 x. \end{aligned}$$

Set $f_2(t):=t^{-(\sigma -1)}$. Then,

$$\begin{aligned} h_2(\sigma )&\asymp -\int ^{+\infty }_{3}A_2(t)f'_2(t)dt +O(1)\asymp (\sigma -1)\int ^{+\infty }_{3}(\log _3 t) t^{-\sigma }dt\\&=(\sigma -1)\left( \int ^{e\sigma '}_{\log 3}+\int ^{+\infty }_{e\sigma '}\right) (\log _2 x )e^{-(\sigma -1)x}dx. \end{aligned}$$

Now

$$\begin{aligned} (\sigma -1)\int ^{e\sigma '}_{\log 3}(\log _2 x) e^{-(\sigma -1)x}dx&\asymp&(\sigma -1)\int ^{e\sigma '}_{\log 3}(\log _2 x) dx\\\le & {} (\sigma -1)e\sigma '\left( \log _2 \left( e\sigma '\right) \right) \lesssim \log _2\sigma '. \end{aligned}$$

We perform a change of variable in the integral over $[e \sigma ',+\infty )$.

$$\begin{aligned} I_{2,2}&:=(\sigma -1)\int ^{+\infty }_{e\sigma '}(\log _2 x) e^{-(\sigma -1)x}dx=\int ^{+\infty }_{e}\left[ \log \left( \log y+\log \sigma '\right) \right] e^{-y}dy\\&\ge (\log _2 \sigma ')\int ^{+\infty }_{e}e^{-y}dy\gtrsim \log _2 \sigma '. \end{aligned}$$

Since $\log (a+b)\le \log a \log b+1, \ \text {for }a\ge e \text { and }b\ge e$, we obtain

$$\begin{aligned} I_{2,2}&\le \int ^{+\infty }_{e}\left[ (\log _2 y)( \log _2\sigma ')+1\right] e^{-y}dy\lesssim \log _2\sigma ', \end{aligned}$$

and $I_{2,2}\asymp \log _2\sigma '$. It follows that $h_2(\sigma )\asymp \log _2\sigma '.$

We now focus on $h_3$. By Mertens’ first Theorem, $A_3(x):= \sum _{3\le p\le x}\frac{\log p}{p}=\log x +O(1),$ and putting $f_3(t):=t^{-(\sigma -1)}\left( \log _2 t\right) ^{-\eta },$ we see that

$$\begin{aligned} h_3(\sigma )&=-\int ^{+\infty }_{3}A_3(t)f'_3(t)dt+O(1)\\&\asymp (\sigma -1)\int ^{+\infty }_{3}\left( \log t \right) t^{-\sigma }\left( \log _2 t\right) ^{-\eta }dt\\&\asymp (\sigma -1)\left( \int ^{\sigma '}_{\log 3}+\int ^{+\infty }_{\sigma '}\right) x e^{- (\sigma -1)x} \left( \log x\right) ^{-\eta }dx. \end{aligned}$$

Integration by parts gives that

$$\begin{aligned} I_{3,1}&:=(\sigma -1)\int ^{\sigma '}_{\log 3}x e^{- (\sigma -1)x} \left( \log x\right) ^{-\eta }dx\\&\asymp (\sigma -1)\int ^{\sigma '}_{\log 3}x \left( \log x\right) ^{-\eta }dx \asymp \sigma '\left( \log \sigma '\right) ^{-\eta }. \end{aligned}$$

Next, (7.4) is a consequence of

$$\begin{aligned} I_{3,2}&:= (\sigma -1)\int ^{+\infty }_{\sigma '}x e^{- (\sigma -1)x} \left( \log x\right) ^{-\eta }dx\\&=\frac{1}{\sigma -1}\int ^{+\infty }_{1}y e^{-y}\left( \log y+\log \sigma '\right) ^{-\eta }dy\\&\lesssim \sigma '\int ^{+\infty }_{1}\frac{y e^{-y}}{\left( \log \sigma '\right) ^{\eta }}dy. \end{aligned}$$

$\square$

Lemma 12

If $2\eta >1$ and $\delta +\eta >1$, we have

$$\begin{aligned} S:= & {} \sum _{p_1,p_2,p_3\in \mathbb {P}, p_j\ge 3}\frac{1}{p_1p_2p_3\left( \log _2 p_1\right) ^{2\delta }\left( \log _2 p_2\right) ^{2}}\times \\&\frac{\left( \log p_3\right) ^2}{ \left( \log _2 p_3\right) ^{2\eta }\left( \log (p_1 p_2 p_3)\right) ^2}<\infty . \end{aligned}$$

Proof

For $p_1,p_2\ge 3,$ we set $L:=\log (p_1p_2)$ and $S_3(p_1,p_2):=\sum _{p_3}\frac{\left( \log p_3\right) ^2}{p_3 \left( \log _2 p_3\right) ^{2\eta }\left( \log p_3+L\right) ^2}.$ Then, we have

$$\begin{aligned} S=\sum _{p_1,p_2,p_3}\frac{1}{p_1p_2\left( \log _2 p_1\right) ^{2\delta }\left( \log _2 p_2\right) ^{2}}S_3(p_1,p_2). \end{aligned}$$

Under the condition $2\eta >1$, the sum $S_3(p_1,p_2)$ converges, and

$$\begin{aligned} S_3(p_1,p_2)&=-\int ^{+\infty }_{3}A_1(t)\frac{d}{dt}\left[ \frac{(\log t)^2}{\left( \log _2 t\right) ^{2\eta }(\log t+L)^2} \right] dt+\frac{O(1)}{L^2}\\&\lesssim \frac{O(1)}{L^2}+\int ^{+\infty }_{3}\frac{\log t}{t(\log _2 t)^{2\eta }(\log t+L)^2}dt\\&=\frac{O(1)}{L^2}+\left( \int ^L_{\log 3}+\int ^{+\infty }_L\right) \frac{xdx}{\left( \log x\right) ^{2\eta }\left( x+L\right) ^2}. \end{aligned}$$

Integration by parts gives

$$\begin{aligned} I_{3,1}&:=\int ^L_{\log 3}\frac{xdx}{\left( \log x\right) ^{2\eta }\left( x+L\right) ^2}\asymp \frac{1}{L^2}\int ^L_{\log 3}\frac{xdx}{\left( \log x\right) ^{2\eta }} \asymp \left( \log L\right) ^{-2\eta }. \end{aligned}$$

We handle the second integral via a change of variable:

$$\begin{aligned} I_{3,2}&:=\int ^{+\infty }_{L}\frac{xdx}{\left( \log x\right) ^{2\eta }\left( x+L\right) ^2} =\left( \int ^{L}_{1}+\int ^{+\infty }_L\right) \frac{ydy}{\left( 1+y\right) ^2\left( \log y+\log L\right) ^{2\eta }}\\&\lesssim \frac{1}{(\log L)^{2\eta }}\int ^{L}_{1}\frac{dy}{y}+\int ^{+\infty }_L\frac{dy}{y(\log y)^{2\eta }}\asymp \left( \log L\right) ^{1-2\eta }. \end{aligned}$$

Therefore

$$\begin{aligned} S_3(p_1,p_2)\lesssim \left( \log L\right) ^{1-2\eta },\ L=\log (p_1p_2). \end{aligned}$$

We next put $M=\log p_1$, and deal with

$$\begin{aligned} S_2(p_1):=\sum _{p_2}\frac{1}{p_2(\log _2 p_2)^2}S_3(p_1,p_2)\lesssim \sum _{p}\frac{1}{p(\log _2 p)^2\left[ \log \left( \log p+M\right) \right] ^{2\eta -1}}. \end{aligned}$$

With the notation $f_2(t):=\left[ {\left( \log _2 t\right) ^2\left[ \log \left( \log t+M\right) \right] ^{2\eta -1}}\right] ^{-1},$ we obtain that

$$\begin{aligned} S_2(p_1)&=\frac{O(1)}{\left( \log M\right) ^{2\eta -1}}-\int ^{+\infty }_{3}A_1(t)f'_2(t)dt \lesssim \frac{O(1)}{\left( \log M\right) ^{2\eta -1}}+I_{2,1}+ I_{2,2}, \end{aligned}$$

where

$$\begin{aligned} I_{2,1}&:= \int ^{+\infty }_{3}\frac{dt}{t\log t\left( \log _2 t\right) ^2\left[ \log \left( \log t+M\right) \right] ^{2\eta -1}} ;\\ I_{2,2}&:= \int ^{+\infty }_{3}\frac{dt}{t\left( \log _2 t\right) \left( \log t+M\right) \left[ \log \left( \log t+M\right) \right] ^{2\eta }}. \end{aligned}$$

We derive

$$\begin{aligned} I_{2,1}&=\left( \int ^{M}_{\log 3} +\int ^{+\infty }_{M}\right) \frac{dx}{x\left( \log x\right) ^2\left[ \log \left( x+M\right) \right] ^{2\eta -1}}\\&\lesssim \frac{1}{\left[ \log M\right] ^{2\eta -1}}\int ^{M}_{\log 3}\frac{dx}{x\left( \log x\right) ^2}\\&\quad +\left( \log M\right) ^{1-2\eta }\int ^{+\infty }_{M}\frac{dx}{x\left( \log x\right) ^2} \lesssim \left( \log M\right) ^{1-2\eta }. \end{aligned}$$

The second integral is estimated in the same way:

$$\begin{aligned} I_{2,2}&=\left( \int ^{M}_{\log 3} +\int ^{+\infty }_{M}\right) \frac{dx}{(x+M)(\log x)\left[ \log (x+M)\right] ^{2\eta }}\\&\lesssim \frac{1}{M(\log M)^{2\eta }}\int ^{M}_{\log 3}\frac{dx}{\log x}+\frac{1}{(\log M)^{2\eta -1}}\int ^{+\infty }_{M}\frac{dx}{x(\log x)^2}\\&\asymp \frac{1}{M(\log M)^{2\eta }}\left( \left[ \frac{x}{\log x}\right] ^{x=M}_{x=\log 3}+\int ^{M}_{\log 3}\frac{x^2}{2}\frac{(\log x)^{-2}}{x}dx\right) \\&\quad + \frac{1}{(\log M)^{2\eta }}\asymp \frac{1}{(\log M)^{2\eta }}. \end{aligned}$$

Therefore, we have

$$\begin{aligned} S_2(p_1)\lesssim \frac{1}{(\log M)^{2\eta -1}},\ M=\log p_1. \end{aligned}$$

It follows that

$$\begin{aligned} S\lesssim \sum _{p_1}\frac{1}{p_1(\log _2 p_1)^{2\delta }}S_2(p_1)\lesssim \sum _{p\ge 3}\frac{1}{p(\log _2 p)^{{\varepsilon }}},\ {\varepsilon }:=2\delta +2\eta -1. \end{aligned}$$

Again, partial summation gives that when ${\varepsilon }>1$,

$$\begin{aligned} \sum _{3\le p}\frac{1}{p(\log _2 p)^{{\varepsilon }}}\asymp {\varepsilon }\int ^{+\infty }_{3}\frac{\log _2 t}{t(\log t)(\log _2 t)^{{\varepsilon }+1}}dt<\infty . \end{aligned}$$

$\square$

Proposition 6

There exists a 3-homogeneous function g which is in $\left( \cap _{0<p<\infty }{{\mathcal {H}}^{p}_{w}}\right) \cap \text {Bloch}_0(\mathbb {C}_{1/2})$, such that $T_g$ is unbounded on ${{\mathcal {H}}^{2}_{w}}$.

Proof

Using Lemma 11, we see that, if $g'=-(h_1h_2h_3)_{\frac{1}{2}}$, $g'$ is convergent on $\mathbb {C}_{1/2}$, and its estimate near the line $\mathfrak {R}s=\frac{1}{2}$ is determined by the behavior of the functions $h_j$ near the line $\mathfrak {R}s=1$. Then g is in $\text {Bloch}_0(\mathbb {C}_{1/2})$, because of

$$\begin{aligned} \left| g'(\sigma )\right| \asymp \frac{1}{\sigma -\frac{1}{2}}\left( \log \frac{1}{\sigma -\frac{1}{2}}\right) ^{1-\delta -\eta }\left( \log _2\frac{1}{\sigma -\frac{1}{2}}\right) ,\ \text { as }\sigma \rightarrow 1/2^+. \end{aligned}$$

On another hand, the 3-homogeneous function

$$\begin{aligned} g(s)=\sum _{n}b_n n^{-s}=\sum _{p_1,p_2,p_3}\frac{\alpha _{1,p_1}\alpha _{2,p_2}\alpha _{3,p_3}}{\log (p_1p_2p_3)}\left( p_1p_2p_3\right) ^{-s} \end{aligned}$$

is in ${{\mathcal {H}}^{2}_{w}}$ by Lemma 12, since $\left\| g\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}} =\sum _{n}\left| b_n\right| ^2 w^{-1}_{n}\asymp \sum _{n}\left| b_n\right| ^2 \asymp S<\infty$.

By Lemma 10, g is in $\cap _{0<p<\infty }{{\mathcal {H}}}^{p}_{w}.$

It remains to prove that $T_g$ is unbounded on ${{\mathcal {H}}^{2}_{w}}$. We again choose as test functions (cf the proof of Proposition 5)

$$\begin{aligned} f_x(s):= \prod _{\frac{x}{2}<p\le x}\left( 1+w^{1/2}_{2}p^{-s}\right) =\sum _{n\ge 1}a_n n^{-s}. \end{aligned}$$

$S_x$ is the set of square free integers generated by $\frac{x}{2}<p\le x$. Set $V_x=\left\{ n\in S_x,\ \omega (n)\ge \frac{N(x)}{2}\right\}$.

For $n\in V_x$, set

$$\begin{aligned} A_n:=\sum _{p_1p_2p_3|n}b_{p_1p_2p_3}\left( \log (p_1p_2p_3)\right) a_{\frac{n}{p_1p_2p_3}} \end{aligned}$$

The coefficients in $A_n$ satisfy

$$\begin{aligned} b_{p_1p_2p_3} \left( \log (p_1p_2p_3)\right) \gtrsim \frac{\log x}{x^{3/2}\left( \log _2 x\right) ^{\eta +\delta +1}}. \end{aligned}$$

Since $\left\| f_x\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}\asymp \left| V_x\right|$, we see that

$$\begin{aligned} \left\| T_g f_x\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}&\ge \sum _{n\in V_x}w^{-1}_{n}\left( \log n\right) ^{-2}A^{2}_{n}\\&\gtrsim \sum _{n\in V_x}w^{-\omega (n)}_{2}\left( \omega (n)\log x\right) ^{-2}\times \\&\quad \left[ \frac{\log x}{x^{3/2}\left( \log _2 x\right) ^{\eta +\delta +1}}\left( {\begin{array}{c}\omega (n)\\ 3\end{array}}\right) \left( w^{1/2}_{2}\right) ^{\omega (n)-3}\right] ^2\\&\gtrsim \left\| f_x\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}} \left( \frac{x}{\log x}\right) ^4 \frac{1}{x^3\left( \log _2 x\right) ^{2(\delta +\eta +1)}}, \end{aligned}$$

and the proof is complete. $\square$

8 Comparison of ${{\mathcal {X}}_{w}}$ with other spaces of Dirichlet series

The previous results enable us to derive some inclusions involving ${{\mathcal {X}}_{w}}$.

In the context of the unit disk, the space of symbols g for which the Volterra operator $J_g$ (1.3) is bounded on $A^2_{\alpha }(\mathbb {D})$ is $\text {Bloch}(\mathbb {D})$. It coincides with the space of holomorphic g such that the Hankel form (1.5) is bounded, and with the dual space of $A^1_{\alpha }(\mathbb {D})$.

We shall study the counterparts of these facts for ${{\mathcal {X}}_{w}}$.

8.1 Bounded Hankel forms

The Hilbert space ${{\mathcal {H}}^{2}_{w}}$ is equipped with the inner product $\left\langle .,.\right\rangle _{{{\mathcal {H}}^{2}_{w}}}$. The Hankel form of symbol $g\in {{\mathcal {D}}}$ is defined on ${{\mathcal {H}}^{2}_{w}}$ by

$$\begin{aligned} H_g(fh):=\left\langle fh,g\right\rangle _{{{\mathcal {H}}^{2}_{w}}}. \end{aligned}$$

(8.1)

We say that $H_g$ is bounded on ${{\mathcal {H}}^{2}_{w}}\times {{\mathcal {H}}^{2}_{w}}$ if there is a constant C such that

$$\begin{aligned} \left| H_g(fh)\right| \le C\left\| f\right\| _{{{\mathcal {H}}^{2}_{w}}}\left\| h\right\| _{{{\mathcal {H}}^{2}_{w}}}\ \text { for }f,h\in {{\mathcal {H}}^{2}_{w}}. \end{aligned}$$

The weak product ${{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}$ is the Banach space defined as the closure of all finite sums $F=\sum _{k}f_k h_k$, where $f_k, h_k\in {{\mathcal {H}}^{2}_{w}}$, under the norm

$$\begin{aligned} \left\| F\right\| _{ {{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}}:=\inf \sum _{k}\left\| f_k\right\| _{{{\mathcal {H}}^{2}_{w}}}\left\| h_k\right\| _{{{\mathcal {H}}^{2}_{w}}} . \end{aligned}$$

Here the infimum is taken over all finite representations of F as $F=\sum _{k}f_k h_k$.

Let ${{\mathcal {Y}}}$ be a Banach space of Dirichlet series in which the space of Dirichlet polynomials ${{\mathcal {P}}}$ is dense. We say that a Dirichlet series $\phi$ is in the dual space ${{\mathcal {Y}}}^{*}$ if the linear functional induced by $\phi$ via the ${{\mathcal {H}}^{2}_{w}}$-pairing is bounded. In other words, $\phi \in {{\mathcal {Y}}}^{*}$ if and only if

$$\begin{aligned} v_{\phi }(f)=\left\langle f,\phi \right\rangle _{{{\mathcal {H}}^{2}_{w}}} ,\ f\in {{\mathcal {P}}}, \end{aligned}$$

extends to a bounded functional on ${{\mathcal {Y}}}$.

From its definition, $H_g$ (8.1) is bounded on ${{\mathcal {H}}^{2}_{w}}$ if and only if $g\in \left( {{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\right) ^{*}$.

We aim to relate Hankel forms and Volterra operators. The primitive of $f\in {{\mathcal {D}}}$ with constant term 0 is denoted by

$$\begin{aligned} {\partial }^{-1}f(s):=-\int ^{+\infty }_{s}f(u)du, \end{aligned}$$

We observe that

$$\begin{aligned} H_g(fh)=f\left( +\infty \right) h\left( +\infty \right) g\left( +\infty \right) +\left\langle {\partial }^{-1}(f'h),g\right\rangle _{{{\mathcal {H}}^{2}_{w}}}+\left\langle {\partial }^{-1}(fh'),g\right\rangle _{{{\mathcal {H}}^{2}_{w}}}. \end{aligned}$$

The Banach space ${\partial }^{-1}\left( {\partial }{{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\right)$ is the completion of the space of Dirichlet series F whose derivatives have a finite sum representation $F'=\sum _{k}f_k h'_k$, under the norm

$$\begin{aligned} \left\| F\right\| _{{\partial }^{-1}\left( {\partial }{{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\right) }:=\left| F(+\infty )\right| +\sum _{k}\left\| f_k\right\| _{{{\mathcal {H}}^{2}_{w}}}\left\| h_k\right\| _{{{\mathcal {H}}^{2}_{w}}}, \end{aligned}$$

where the infimum is taken over all finite representations. The product rule $(fg)'=f'g+fg'$ implies that

$$\begin{aligned} {{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\subset {\partial }^{-1}\left( {\partial }{{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\right) , \end{aligned}$$

and then

$$\begin{aligned} \left( {\partial }^{-1}\left( {\partial }{{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\right) \right) ^{*} \subset \left( {{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\right) ^{*}. \end{aligned}$$

(8.2)

It has been shown in [14] that, for the space ${{\mathcal {H}}}^2_{0}=\left\{ f\in {{\mathcal {H}}}^2\ :\ f(+\infty )=0\right\}$, the inclusion $\left( {\partial }^{-1}\left( {\partial }{{\mathcal {H}}}^2_{0}\odot {{\mathcal {H}}}^2_{0} \right) \right) ^{*} \subset \left( {{\mathcal {H}}}^2_{0}\odot {{\mathcal {H}}}^2_{0}\right) ^{*}$ is strict. As for the space ${{\mathcal {H}}^{2}_{w}}$, the question whether the inclusion is strict remains open.

The membership of g in $\left( {\partial }^{-1}\left( {\partial }{{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\right) \right) ^{*}$ is equivalent to the boundedness of the so-called “half-Hankel form”

$$\begin{aligned} (f,h)\mapsto \left\langle {\partial }^{-1}(f'h),g\right\rangle _{{{\mathcal {H}}^{2}_{w}}}. \end{aligned}$$

(8.3)

As in the case of ${{\mathcal {H}}}^2$, the boundedness of $T_g$ implies the boundedness of $H_g$.

Theorem 5

If the Volterra operator $T_g$ is bounded on ${{\mathcal {H}}^{2}_{w}}$, then the Hankel form $H_g$ is bounded.

Proof

We adapt the proof of Corollary 6.2 in [13] to the framework of the polydisk $\mathbb {D}^{\infty }$. Polarizing the Littlewood–Paley formula (1), we get

$$\begin{aligned} \left\langle f,g\right\rangle _{{{\mathcal {H}}^{2}_{w}}}= & {} f(+\infty ) g(+\infty )+4\int _{\mathbb {D}^{\infty }}\int _{\mathbb {R}}\int ^{+\infty }_{0}f'_{\chi }(\sigma +it)\overline{g'_{\chi }(\sigma +it)}\sigma d\sigma \frac{dt}{1+t^2}d\mu _w(\chi ). \end{aligned}$$

Then, we derive an expression of the half-Hankel form

$$\begin{aligned} \left\langle {\partial }^{-1}(f'h),g\right\rangle _{{{\mathcal {H}}^{2}_{w}}}= & {} 4 \int _{\mathbb {D}^{\infty }}\int _{\mathbb {R}}\int ^{+\infty }_{0}f'_{\chi }(\sigma +it)h_{\chi }(\sigma +it)\overline{g'_{\chi }(\sigma +it)}\sigma d\sigma \frac{dt}{1+t^2}d\mu _w(\chi ). \end{aligned}$$

Since $T_g$ is bounded on ${{\mathcal {H}}^{2}_{w}}$, the Carleson measure characterization (4.1) induces that the form (8.3) is also bounded. Then $H_g$ is bounded on ${{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}$ by the inclusion (8.2). $\square$

The previous Theorem states that we have

$$\begin{aligned} {{\mathcal {X}}_{w}}\subset \left( {{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\right) ^{*}. \end{aligned}$$

The rest of the section is devoted to study the reverse inclusion.

Let $l^{2}_{w}$ denote the Hilbert space of complex sequences $a=(a_n)_n$ such that

$$\begin{aligned} \left\| a\right\| _{l^{2}_{w}} :=\left( \sum _{n\ge 1}\frac{\left| a_n\right| ^2}{w_n}\right) ^{1/2} <\infty . \end{aligned}$$

A sequence $(\rho _n)_n$ generates the following multiplicative Hankel form

$$\begin{aligned} \rho (a,b):=\sum ^{+\infty }_{n=1}\sum ^{+\infty }_{m=1}a_mb_n\frac{\rho _{mn}}{w_{mn}},\ a,b\in l^{2}_{w}. \end{aligned}$$

(8.4)

The symbol of the form is the Dirichlet series $g(s)=\sum _{n\ge 1}\overline{\rho _n} n^{-s}$. The form $\rho$ is said to be bounded if there is a constant C such that

$$\begin{aligned} \left| \rho (a,b)\right| \le C\left\| a\right\| _{l^{2}_{w}} \left\| b\right\| _{l^{2}_{w}}. \end{aligned}$$

If f and h are Dirichlet series with coefficients a and b, respectively, we have

$$\begin{aligned} H_{g}(fh)=\left\langle fh,g\right\rangle _{{{\mathcal {H}}^{2}_{w}}}= \rho (a,b). \end{aligned}$$

When the symbol g has non negative coefficients, there is equivalence between the boundedness of $H_g$ and the half-Hankel form (8.3). In fact, the proof given for ${{\mathcal {H}}}^2$ in [14] is valid for the spaces ${{\mathcal {H}}^{2}_{w}}$.

Proposition 7

Let $g(s)=\sum _{n\ge 1}\overline{\rho _n} n^{-s}$ be in ${{\mathcal {H}}^{2}_{w}}$. The linear functional defined on ${{\mathcal {H}}^{2}_{w}}$

$$\begin{aligned} v_g(f) :=\left\langle f,g\right\rangle _{{{\mathcal {H}}^{2}_{w}}} \end{aligned}$$

is bounded on ${\partial }^{-1}\left( {\partial }{{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\right)$ if and only if the weighted form

$$\begin{aligned} J_g(a,b)=\sum ^{+\infty }_{n=1}\sum ^{+\infty }_{m=1}a_m b_n\frac{\log n}{\log m+ \log n}\frac{\rho _{mn}}{w_{mn}}, \end{aligned}$$

(where it is understood that for $m=n=1$, the summand is 0) is bounded on $l^{2}_{w}\odot l^{2}_{w}$. The norms are equivalent, i.e.

$$\begin{aligned} \left\| g\right\| _{\left( {\partial }^{-1}\left( {\partial }{{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\right) \right) ^{*}}\asymp \left\| v_g\right\| \asymp \left| \rho _1\right| +\left\| J_g\right\| . \end{aligned}$$

If $\rho _k\ge 0$ for all k, then $g\in \left( {\partial }^{-1}\left( {\partial }{{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\right) \right) ^{*}$ if and only if $g\in \left( {{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\right) ^{*}$, with equivalent norms.

Proposition 7 will enable us to provide examples of symbols g for which the Hankel form $H_g$ and the half-Hankel form (8.3) are bounded, but the Volterra operator $T_g$ is unbounded (see the proof of Proposition 9). This differs from the case of weighted Dirichlet spaces on the unit disk, for which the boundedness of $H_g$, the form (8.3) and $T_g$ are equivalent [1].

For convergence reasons, we will consider Hankel forms defined on Dirichlet series without constant term. So we will work on the space

$$\begin{aligned} {{\mathcal {H}}}^2_{w,0}=\left\{ f\in {{\mathcal {H}}}^2_w\ :\ f(+\infty )=0\right\} . \end{aligned}$$

We have seen in Lemma 1 that the space ${{\mathcal {H}}^{2}_{w}}$ is embedded in a Bergman space of the form $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$. For $\delta >0$, it is thus natural to define the Hankel form

$$\begin{aligned} H^{\left( \delta \right) }(fh):=\int ^{+\infty }_{1/2}f(\sigma )h(\sigma )\left( \sigma -\frac{1}{2}\right) ^{\delta }d\sigma ,\ f,h\in {{\mathcal {H}}^{2}_{w}}_{,0}. \end{aligned}$$

(8.5)

Such multiplicative forms have been considered in the context of ${{\mathcal {H}}}^2$ [12] and on ${{\mathcal {A}}}^{2}_{1}$ [9].

Since $K^{{{\mathcal {H}}^{2}_{w}}}(s,u)-1=\sum _{n\ge 2}w_n n^{-\overline{u}}n^{-s}$ is the reproducing kernel of ${{\mathcal {H}}^{2}_{w}}_{,0}$, we see that $H^{\left( \delta \right) }(fh)=\left\langle fh,\phi _{\delta }\right\rangle _{{{\mathcal {H}}^{2}_{w}}}$, where

$$\begin{aligned} \phi _{\delta }(s)=\int ^{+\infty }_{1/2}\left[ K^{{{\mathcal {H}}^{2}_{w}}}(s, \sigma )-1\right] \left( \sigma -\frac{1}{2}\right) ^{\delta }d\sigma =\sum ^{+\infty }_{n=2}\frac{w_n}{\sqrt{n}\left( \log n\right) ^{\delta +1}}n^{-s}. \end{aligned}$$

Proposition 8

Let $\delta >0$ as in (2.5). Then $H^{\left( \delta \right) }$ defined in (8.5) is a multiplicative Hankel form with symbol $\phi _{\delta }$, which is bounded on ${{\mathcal {H}}^{2}_{w}}_{,0}\odot {{\mathcal {H}}^{2}_{w}}_{,0}$.

Proof

The proof is similar to that of Theorem 13 in [9]. The Cauchy-Schwarz inequality ensures that

$$\begin{aligned} \left| H^{\left( \delta \right) }(fh)\right| \le \left( \int ^{+\infty }_{1/2}\left| f(\sigma )\right| ^2\left( \sigma -\frac{1}{2}\right) ^{\delta }d\sigma \right) ^{1/2} \left( \int ^{+\infty }_{1/2}\left| h(\sigma )\right| ^2\left( \sigma -\frac{1}{2}\right) ^{\delta }d\sigma \right) ^{1/2}. \end{aligned}$$

If $f(s)=\sum ^{+\infty }_{n=2}a_n n^{-s}$, notice the pointwise estimate

$$\begin{aligned} \left| f(\sigma )\right| ^2\le \left\| f\right\| ^2_{{{\mathcal {H}}^{2}_{w}}} \left( \sum ^{+\infty }_{n=2}w_n n^{-2\sigma }\right) \lesssim \left\| f\right\| ^2_{{{\mathcal {H}}^{2}_{w}}}4^{-\sigma },\ \text { for }\sigma \ge 1. \end{aligned}$$

Since the bounded measure $d\mu (\sigma +it)=\chi _{)1/2, 1]}(\sigma )\left( \sigma -\frac{1}{2}\right) ^{\delta }d\sigma$, supported on the real line, is Carleson for $A_{i,\delta }\left( \mathbb {C}_{1/2}\right)$, $\mu$ is Carleson for ${{\mathcal {H}}^{2}_{w}}$ by Lemma 6, and

$$\begin{aligned} \int ^{+\infty }_{1/2}\left| f(\sigma )\right| ^2\left( \sigma -\frac{1}{2}\right) ^{\delta }d\sigma&=\left( \int ^{1}_{1/2}+\int ^{+\infty }_{1}\right) \left| f(\sigma )\right| ^2\left( \sigma -\frac{1}{2}\right) ^{\delta }d\sigma \lesssim \left\| f\right\| ^2_{{{\mathcal {H}}^{2}_{w}}}. \end{aligned}$$

$\square$

We next exhibit symbols giving rise to bounded Hankel forms and bounded half-Hankel forms, though the associated Volterra operator is unbounded.

Proposition 9

We have the strict inclusions

$$\begin{aligned} {{\mathcal {X}}}({{\mathcal {H}}^{2}_{w}}_{,0})&\subset _{\ne }\left( {{\mathcal {H}}^{2}_{w}}_{,0}\odot {{\mathcal {H}}^{2}_{w}}_{,0}\right) ^{*};\\ {{\mathcal {X}}_{w}}&\subset _{\ne }\left( {{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\right) ^{*}. \end{aligned}$$

Proof

It just remains to check the strictness of the inclusions. For the exponent $\delta =\delta (w)$ and $\frac{1}{2}\le a<1$, consider the symbol in ${{\mathcal {H}}^{2}_{w}}_{,0}$

$$\begin{aligned} g(s)=\sum ^{+\infty }_{n=2}\frac{w_n}{{n}^a\left( \log n\right) ^{\delta +1}}n^{-s}. \end{aligned}$$

From Proposition 8 and the fact that the coefficients are positive, g is in $\left( {{\mathcal {H}}^{2}_{w}}_{,0}\otimes {{\mathcal {H}}^{2}_{w}}_{,0}\right) ^{*}$ for any $\frac{1}{2}\le a<1$. In fact, the half Hankel form corresponding to g is bounded. We have seen in Proposition 4 that $T_g$ is not bounded on ${{\mathcal {H}}^{2}_{w}}.$ Since $T_g 1=g$, g does not belong to ${{\mathcal {X}}}({{\mathcal {H}}^{2}_{w}}_{,0})$.

In order to prove that $g\in \left( {{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}\right) ^{*}$, we consider the associated multiplicative form $\rho$ (8.4). Let f, h be Dirichlet series with coefficients a, b, belonging to ${{\mathcal {H}}^{2}_{w}}$. Since

$$\begin{aligned} \rho (a,b)&=\sum _{m,n\ge 2}a_m b_n\frac{\rho _{mn}}{w_{mn}}+a_1\sum ^{+\infty }_{n=1}b_n\frac{\rho _{n}}{w_{n}}+b_1\sum ^{+\infty }_{m=1}a_m\frac{\rho _{m}}{w_{m}}\\&=H_g\left( \left( f-f\left( \infty \right) \right) \left( g-g\left( \infty \right) \right) \right) +f\left( \infty \right) \left\langle h,g\right\rangle _{{{\mathcal {H}}^{2}_{w}}}+g\left( \infty \right) \left\langle f,g\right\rangle _{{{\mathcal {H}}^{2}_{w}}}, \end{aligned}$$

the first part of the proof entails that $H_g$ is bounded on ${{\mathcal {H}}^{2}_{w}}\odot {{\mathcal {H}}^{2}_{w}}$. $\square$

8.2 ${{\mathcal {X}}_{w}}$ and the dual of ${{\mathcal {H}}}^{1}_{w}$

Keeping in mind the results known for Bergman spaces of the unit disk, it is natural to compare ${{\mathcal {X}}_{w}}$ and $\left( {{\mathcal {H}}}^{1}_{w}\right) ^{*}$.

In general, the dual of ${{\mathcal {H}}}^{1}_{w}$ is not known. However, it is shown in [9] that

$$\begin{aligned} {{\mathcal {K}}}\subset \left( {{\mathcal {A}}}^{1}_{1}\right) ^{*}, \end{aligned}$$

(8.6)

where ${{\mathcal {K}}}$ is the space of Dirichlet series $f(s)=\sum ^{+\infty }_{n=1}a_n n^{-s}$ such that

$$\begin{aligned} \sum ^{+\infty }_{n=1}\frac{d_4(n)}{\left[ d(n)\right] ^2}\left| a_n\right| ^2<\infty . \end{aligned}$$

The following consequence of this inclusion will stress upon the difference between the finite and infinite dimensional setting.

Proposition 10

$\left( {{\mathcal {A}}}^{1}_{1}\right) ^{*}$ is not contained in ${{\mathcal {X}}}\left( {{\mathcal {A}}}^{2}_{1}\right)$.

Proof

By Abel summation and the Chebyshev estimate, the symbol

$$\begin{aligned} g(s)=\sum ^{+\infty }_{n=2}\frac{d(n)}{n^{a}(\log n)^2} n^{-s},\text { for }\frac{1}{2}< a <1, \end{aligned}$$

is in ${{\mathcal {K}}}$, and thus in $\left( {{\mathcal {A}}}^{1}_{1}\right) ^{*}$. However, $T_g$ is unbounded on ${{\mathcal {A}}}^{2}_{1}$ (Proposition 4). $\square$

8.3 ${{\mathcal {X}}_{w}}$ and the spaces ${{\mathcal {H}}}^{p}_{w}$

It has been shown in [13] that $BMOA(\mathbb {C}_0)\cap {{\mathcal {D}}}\subset _{\ne } {{\mathcal {X}}}({{\mathcal {H}}}^2) \subset _{\ne }\cap _{0<p<\infty }{{\mathcal {H}}}^{p}.$ We have an analogue for Bergman spaces of Dirichlet series.

Theorem 6

We have the strict inclusions

$$\begin{aligned} BMOA(\mathbb {C}_0)\cap {{\mathcal {D}}}\subset _{\ne } {{\mathcal {X}}_{w}}\subset _{\ne }\cap _{0<p<\infty }{{\mathcal {H}}}^{p}_{w}. \end{aligned}$$

Proof

The inclusions have been proved in Theorem 1 and Corollary 1. As observed in [13], the symbols $g(s)=\sum ^{+\infty }_{n=2}\frac{\psi (n)}{\log n}n^{-s}$, where $\psi$ is the completely multiplicative function defined on the primes by $\psi (p):=\lambda p^{-1}\log p$, $0<\lambda \le 1$, are in ${{\mathcal {X}}}({{\mathcal {H}}}^2)$, and satisfy

$$\begin{aligned} \sum ^{+\infty } _{n=1}\psi (n)n^{-\sigma }\asymp \exp \left( \lambda \sum _p\frac{\log p}{p^{1+\sigma }}\right) \asymp \exp \left( \lambda \frac{1}{\sigma }\right) , \ \sigma >0. \end{aligned}$$

Hence, they are not in $BMOA(\mathbb {C}_0)$, though they belong to ${{\mathcal {X}}_{w}}$ (Lemma 9).

The second inclusion is strict by Proposition 6. $\square$

With the method of Proposition 4, one can show that $g(s)=\sum _{n\ge 2}\frac{n^{-a}}{\log n}n^{-s}$, $1/2\le a<1$, is not in ${{\mathcal {X}}_{w}}$, though it belongs to $BMOA(\mathbb {C}_{1-a})$ [13]. Therefore, we have the strict inclusion

$$\begin{aligned} {{\mathcal {X}}_{w}}\subset _{\ne } \text {Bloch}(\mathbb {C}_{1/2}). \end{aligned}$$

8.4 ${{\mathcal {X}}_{w}}\cap {{\mathcal {D}}}_{d}$ and Bloch spaces

Theorem 7

Let d be a positive integer. The following inclusions hold

$$\begin{aligned} {{\mathcal {D}}}_{d}\cap \text {Bloch}(\mathbb {C}_0)\subset {{\mathcal {D}}}_{d}\cap {{\mathcal {X}}_{w}}\subset _{\ne }{{\mathcal {B}}}^{-1}\text {Bloch}(\mathbb {D}^d). \end{aligned}$$

Proof

The first inclusion has been shown in Theorem 1(a).

If g is in ${{\mathcal {D}}}_{d}\cap {{\mathcal {X}}_{w}}$, Theorem 5 implies that $H_g$ is bounded on ${{\mathcal {H}}^{2}_{w}}$. Therefore, the form $H_{{{\mathcal {B}}}g}$ (1.4) is bounded on the Bergman space $H^{2}_{w}(\mathbb {D}^d)$. From [17], ${{\mathcal {B}}}g$ is in $\text {Bloch}(\mathbb {D}^d).$

Here is a function g which is not in ${{\mathcal {X}}_{w}}$, such that ${{\mathcal {B}}}g$ is in $\text {Bloch}(\mathbb {D}^2).$ Suppose that

$$\begin{aligned} g'(s)= \frac{1}{1-2^{-s}}\log \left( \frac{1}{1-3^{-s}}\right) ,\ s\in \mathbb {C}_0. \end{aligned}$$

Straightforward computations show that ${{\mathcal {B}}}g\in \text {Bloch}(\mathbb {D}^2).$ The norms $\left\| .\right\| _{A^{2}_{\beta }(\mathbb {D}^2)}$ and $\left\| .\right\| _{B^{2}_{\beta }(\mathbb {D}^2)}$ being equivalent, our setting will be the space $A^{2}_{\beta }(\mathbb {D}^2)$. Now, for

$$\begin{aligned} F(z)=\sum ^{\infty }_{n=1}\frac{\left( n+1\right) ^{\frac{\beta -1}{2}}}{\log (n+1)}z^n=\sum ^{\infty }_{n=0}a_n z^n,\ z\in \mathbb {D}, \end{aligned}$$

define $f(s)=F(2^{-s}) F(3^{-s})$, for $s\in \mathbb {C}_0$. We have

$$\begin{aligned} \left\| f\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}= \left\| F\right\| ^{4}_{A^{2}_{\beta }(\mathbb {D})}\asymp \left( \sum ^{\infty }_{n=1}\frac{1}{\left( n+1\right) \left( \log (n+1)\right) ^2}\right) ^2<\infty . \end{aligned}$$

Putting

$$\begin{aligned} h_1(z_1)&=F(z_1)\frac{1}{1-z_1}=\sum ^{\infty }_{m=0}A_m z^{m}_{1},\ z_1\in \mathbb {D},\\ h_2(z_2)&=F(z_2)\log \left( \frac{1}{1-z_2}\right) =\sum ^{\infty }_{n=0}B_n z^{n}_{2},\ z_2\in \mathbb {D}, \end{aligned}$$

we have $A_m \gtrsim \frac{\left( m+1\right) ^{\frac{\beta +1}{2}}}{\log (m+1)}$ and $B_n\gtrsim \left( n+1\right) ^{\frac{\beta -1}{2}}$. Therefore,

$$\begin{aligned} \left\| T_g f\right\| ^{2}_{{{\mathcal {H}}^{2}_{w}}}&=\left\| R^{-1}\left( h_1 h_2\right) \right\| ^{2}_{A^{2}_{\beta }(\mathbb {D}^2)}\asymp \sum _{m,n\ge 1}\frac{\left| A_m\right| ^2 \left| B_n\right| ^2}{(m+n+1)^2(m+1)^{\beta } (n+1)^{\beta }}\\&\gtrsim \sum _{m\ge 1} \frac{m+1}{\left( \log (m+1)\right) ^2}\frac{\log (m+1)}{(m+1)^2}=\sum _{m\ge 1}\frac{1}{(m+1)\log (m+1)}=+\infty , \end{aligned}$$

which proves the claim. $\square$

A consequence of Theorems 1 and 6 is that

$$\begin{aligned} \text {Bloch}(\mathbb {C}_0)\cap {{\mathcal {D}}}_d\subset \cap _{0<p<\infty }{{\mathcal {H}}}^{p}_{d,w}. \end{aligned}$$

This inclusion can be viewed as a counterpart of the situation of the disk, where $\text {Bloch}(\mathbb {D})\subset \cap _{0<p<\infty }A^{p}_{\beta }(\mathbb {D}).$