Volterra operators and Hankel forms on Bergman spaces of Dirichlet series

For a Dirichlet series g, we study the Volterra operator Tg of symbol g, acting on a class of weighted Hilbert spaces of Dirichlet series. We obtain sufficient / necessary conditions for Tg to be bounded (resp. compact), involving BMO and Bloch type spaces on some half-plane. We also investigate the membership of Tg in Schatten classes. We also relate the boundedness of Tg to the boundedness of a multiplicative Hankel form of symbol g.

For a real number θ, C θ stands for the half-plane {s, ℜs > θ}, and D for the unit disk.D denotes the class of functions f of the form (1.1) in some half-plane C θ , and P is the space of Dirichlet polynomials.
The increasing sequence of prime numbers will be denoted by (pj) j≥1 , and the set of all primes by P. Given a positive integer n, n = p κ will stand for the prime number factorization n d , which associates uniquely to n the finite multi-index κ(n) = (κ1, κ2, • • • , κ d ).The number of prime factors in n is denoted by Ω(n) (counting multiplicities), and by ω(n) (without multiplicities).
The space of eventually zero complex sequences c00 consists in all sequences which have only finitely many non zero elements.We set D ∞ fin = D ∞ ∩ c00 and N ∞ 0,fin = N ∞ 0 ∩ c00, where N0 = N ∪ {0} is the set of non-negative integers.
Let F : D ∞ fin → C be analytic, i.e. analytic at every point z ∈ D ∞ fin separately with respect to each variable.Then F can be written as a convergent Taylor series The truncation AmF of F onto the first m variables is defined by For z, χ in D ∞ , we set z.χ := (z1χ1, z2χ2, • • • ), and p x := (p x 1 , p x 2 , • • • ) for a real number x, .The Bohr lift [12] of the Dirichlet series f (s) = +∞ n=1 ann −s is the power series with the multiindex notation χ α = χ α 1 1 χ α 2 2 • • • .Given a sequence of positive numbers w = (wn)n = (w(n)) n , one considers the Hilbert space (see [25,23]) The choice wn = 1 corresponds to the space H 2 , introduced in [20].
2010 Mathematics Subject Classification.Primary 31B10 and 32A36.Secondary 30B50 and 30H20.Key words and phrases.Volterra operator, Dirichlet series, Hankel forms.The author was supported by the FWF project P 30251-N35.
On the unit disk D, the Volterra operator, whose symbol is an analytic function g, is given by Pommerenke [29] showed that Jg (1.3) is bounded on the Hardy space H 2 (D) if and only if g is in BM OA(D).Let σ be the Haar measure on the unit circle T. Fefferman's duality Theorem states that BM OA(D) is the dual space of H 1 (D).Thus the boundedness of Jg is equivalent to the boundedness of the Hankel form (1.4) Let V be the Lebesgue measure on C, normalized such that V (D) = 1.Many authors, in particular [2], have studied Volterra operators on Bergman spaces of D. The classical Bergman space A 2 γ (D), γ > 0, is associated to the measure d mγ(z) := γ 1 − |z| 2 γ−1 dV (z).Jg is bounded on A 2 γ (D) if and only if g is in the Bloch space, which is the dual of A 1 γ (D).
The Bergman space of the finite polydisk A 2 γ (D d ), d ≥ 1, corresponds to the measure The boundedness of the Hankel form (1.5) Hg(f, h) := is equivalent to the membership of g to the Bloch space (see [18]), defined by Recall that for 0 < p < ∞, the Hardy space of Dirichlet series H p is the space of Dirichlet series f ∈ D such that Bf is in H p (D ∞ ), endowed with the norm , σ∞ being the Haar measure of the infinite polytorus T ∞ .
The norm in the space Let H ∞ (D ∞ ) be the space of series F which are finitely bounded, i.e.
Via the Bohr isomorphism, we have [17,20] (1.6) Several abscissae are related to a function g in D, of the form g(s) = +∞ n=1 bnn −s : the abscissa of convergence σc = inf σ ∈ R : The abscissa of regularity and boundedness, denoted by σ b , is the infimum of those θ such that g(s) has a bounded analytic continuation, to the half-plane ℜ(s) > θ + ǫ, for every ǫ > 0.
We have −∞ ≤ σc ≤ σu ≤ σa ≤ +∞, and, if any of the abscissae is finite σa − σc ≤ 1.Moreover, it is known that σ b = σu [12] , and σa − σu ≤ 1 2 .Volterra operators (1.2) on the spaces H p have been investigated in [14].Our aim is to study similar questions for the spaces H 2 w , associated to specific weights w in the class W defined below.
As in the case of H 2 [14], we obtain sufficient / necessary conditions for Tg to be bounded on the Hilbert spaces H 2 w .However, due to the lack of information of the behavior of the symbols in the strip 0 < ℜs < 1/2, it seems difficult to get an " if and only if" condition.In the Hardy space setting, it is shown that Tg is bounded on H 2 provided that g in BM OA(C0).Since the spaces A 2 β and B 2 β (see section 2) locally behave like Bergman spaces of the half plane C0, we would expect that the membership of g in Bloch(C0) (resp.Bloch0(C0)) would imply the boundedness (resp.compactness) of Tg on H 2 w .We obtain such a sufficient condition when Bg depends on a finite number of variables z1, • • • , z d .However, our method specfically uses that d is finite, and we do not know whether the same result holds if Bg is a function of infinitely many variables.
Le N d be the set of positive integers which are multiples of the primes p1, • • • , p d , One of our main results is the following.
Theorem 1.Let Tg be the operator defined by (1.2) for some Dirichlet series g in D.
Via the Bohr lift, H 2 w are L 2 -spaces of functions on the polydisk D ∞ .Precisely, there exists a probability measure µw on D ∞ such that Analogously to the spaces H p , we define the space H p w , 0 < p < ∞ (see Section 2), as the closure of Dirichlet polynomials under the norm (quasi-norm if 0 < p < 1) w ) be the space of symbols g giving rise to bounded operators Tg on H 2 w .Our study provides the following strict inclusions: We will also compare Xw with other spaces of Dirichlet series, in particular with the dual of H 1 w , and the space of symbols g generating a bounded Hankel form on the weak product H 2 w ⊙ H 2 w .As in the case of H 2 [14], we only get partial results.For Dirichlet series involving d primes, we have The paper is organized as follows.Section 2 starts by presenting some properties of the spaces H 2 w .As a space of analytic functions on the half-plane C 1/2 , H 2 w is continuously embedded in a space of Bergman type of C 1/2 .In view of the Bohr lift, the norm of H 2 w can be expressed in terms of a probability measure µw on the polydisk.For 0 < p < ∞, we consider the Bohr-bergman space H p w , and derive equivalent norms for these spaces.
In section 3, we present some properties of the Dirichlet series which belong to a BMO or Bloch space of some half-plane C θ .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 To prove (b), we observe that the boundedness of Tg on H 2 implies the boundedness of Tg on H 2 w .On another hand, combining the fact that H 2 w is embedded in a Bergman type space of the half-plane C 1/2 with some characterizations of Carleson measures, we establish that Compactness and Schatten classes are considered in Sections 5 and 6.
In section 7, we consider some specific symbols: fractional primitives of translates of a "weighted zeta"-function and homogeneous symbols.These examples will be used in section 8.
In Section 8, we investigate the relationship between the boundedness of the Volterra operator Tg, the boundedness of the Hankel form , and the membership of g in the dual of 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 [13].Additionally, we show the strictness of the inclusions derived previously BM OA(C0) ∩ D ⊂ = Xw ⊂ = ∩0<p<∞H p w , and compare the space D d ∩ Xw with Bloch spaces.
For two functions f, g, the notation f = O(g) or f g, means that there exists a constant C such that f ≤ Cg .If f = O(g) and g = O(f ), we write f ≍ g.

THE BOHR-BERGMAN
β .These spaces are related to number theory.The number of divisors of the integer n, d(n), is d(n) = (κ1 +1) • • • (κ d +1) when n = p κ .We consider the following scale of Hilbert spaces The case β = 0 corresponds to the Hardy space H 2 .The reproducing kernels of B 2 β are It is shown in [34] that there exists φ β (s), an Euler product which converges absolutely in Another family of spaces arises from the so-called generalized divisor function.For γ > 0, the numbers dγ(n) are defined by the relation A computation involving Euler products shows that we have r! , for p ∈ P, and any integer r.
From its definition, dγ is a multiplicative function, i.e. dγ (kl) = dγ (k)dγ (l) if k and l are relatively prime.Thus, dγ (n) can be computed explicitly from the decomposition n = p κ .
We define the spaces Notice that, in each case, the reproducing kernel has the form where Zw(s) := +∞ n=1 wnn −s has a singularity at s = 1, with an estimate of the type (2.7) 2.2.Bohr-Bergman spaces on D ∞ .The Bohr correspondence is an isometry between H 2 w and the weighted Bergman space of the infinite polydisk , where wν = j wν j .
In particular, the space H 2 is identified with the Hardy space H 2 (T ∞ ) [20].
Let us consider the following probability measures on the unit disk D, yields that, the spaces B 2 β (D d ) and A 2 β (D d ) coincide as sets, with equivalent norms.However, the norms are no longer equivalent in the case of infinitely many variables.
The H 2 w -norm will be computed via the rotation invariant probability measure Applying the Bohr lift to a Dirichlet series f (s) = +∞ n=1 ann −s , and using (2.8) for each variable, one obtains the following formula (see [5] in the case of B 2 β ) Definition 2. For 0 < p < ∞, the Bohr-Bergman spaces of Dirichlet series B p β and A p β -denoted by H p ware the completions of the Dirichlet polynomials in the norm (quasi norm when 0 < p < 1) The Kronecker flow of the point χ = (χ1, χ2, which defines an ergodic flow on T ∞ by Kronecker's theorem. Therefore, it follows from Fubini's Theorem that, for any rotation invariant probability measure dν on D ∞ and any probability measure dλ on R, we have (2.9) 2.3.On the half-plane C 1/2 .For θ ∈ R, let τ θ be the following mapping from D to C θ , (2.10) For δ > 0, the conformally invariant Bergman space A i,δ C 1/2 is the space of those functions f which are analytic in C 1/2 , and such that The weights w of the class W satisfy a Chebyshev-type estimate (2.11) For any real number τ , set As mentioned in the introduction, the Dirichlet series which belong the H 2 w absolutely converge in C 1/2 .The space H 2 w is locally embedded in A i,δ(w) C 1/2 [25,27], which means

Since functions in H 2
w are uniformly bounded in C1, these embeddings are global (see [5,9]).
Recall that τ θ , θ ∈ R, is the conformal mapping defined in (2.10).For 0 < p < ∞, the conformally invariant Hardy space H p i (C θ ), is the space of those functions f such that f •τ θ is in H p (T), the usual Hardy space of the unit disk.Setting Let f be in H p w .In view of relation (2.9), and using the same argument as in [6,20], one can prove that for almost all χ, with respect to µw , fχ can be extended analytically on C0 to an element of H p i (C0).The norm of f in H p w can be expressed as

2.5.
A Littlewood-Paley formula.We now derive another expression for the norm in H p w .
Proposition 1.Let λ be a probability measure on R, and p ≥ 1.
When p = 2, we have f 2 Proof.Since the real variable t corresponds to a rotation in each variable of D ∞ , the rotation invariance of µw entails that Ip(f ) does not depend on the choice of the probability measure λ.For general p ≥ 1, we prove (a), by using (2.14).We adapt the argument from [11] (for H p ), by integrating over the polydisk D ∞ instead of the polytorus T ∞ .Suppose f is in H 2 w , and take y > 0. From (2.9) and the rotation invariance, we obtain Integration against y on (0, +∞) gives the formula (see details in [8] for the case of where f * (s) := +∞ n=1 |an| n −s is bounded on C2.Integrating on D ∞ with respect to µw, and using (2.9), we get that The martingale (AmBf )m (with respect to the increasing sequence of σ-algebras of the sets D m × {0}) converges in L p (D ∞ , µw) to Bf .Polynomial approximation in the Bergman spaces of the finite polydisks D m shows that Bf is in BH p w .

SPACES OF SYMBOLS OF VOLTERRA OPERATORS IN HALF-PLANES
If g is in D, the definition (1.2) of Tg shows that we can assume that g (+∞) = 0, i.e.
As in the study of Volterra operators on Bergman spaces the unit disk [2], and on the space of Dirichlet series H 2 [14], the boundedness of Tg on 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 H 2 w or the Bergman space The smallest such constant, denoted by µ CM (Y ) , is called the Carleson constant for µ with respect to Y .A Carleson measure µ is a vanishing Carleson measure for Y if we have for every weakly compact sequence 3.1.BM O spaces of Dirichlet series.The space BM OA(C θ ) consists of holomorphic functions g in the half-plane C θ which satisfy Any g in D ∩ BM OA(C0) has an abscissa of boundedness σ b ≤ 0 (Lemma 2.1 of [14]).
The space V M OA(C0) consists in those functions g in BM OA(C0) such that 3.2.Bloch spaces of Dirichlet series.The Bloch space Bloch(C θ ) consists of holomorphic functions in the half-plane C θ which satisfy It follows that g ′ , and then g is bounded in Cǫ; (a) is proved.Now fix σ > 0. Let m ≥ 1 be an integer, and z = (z1, From the properties of H ∞ and the proof of (a), we have and If 0 < δ < y0 − 1 2 , the Cauchy-Schwarz inequality and Parseval's relation induce that . We now get (c) from the chain of inequalities Now, recall severals of Bloch functions, which are extracted from [2,19].
Lemma 4. Assume δ > 0. For g holomorphic in C θ , the following are equivalent: (e) The operator J h , given by are comparable.

3.3.
Carleson measures on the half-plane C 1/2 .On C 1/2 , we consider Carleson squares is the midpoint of the right edge of the square and ǫ = σ0 − 1 2 .We need the following property (see section 7.2 in [35]).Lemma 5. Let δ > 0 and let µ be a Borel measure on C 1/2 .Then µ is a Carleson measure for A i,δ C 1/2 if and only if, for every square Q(s0), with s0 = σ0 + it0, we have In addition, µ is a vanishing Carleson measure for A i,δ C 1/2 if and only if, uniformly for t0 in R, By Lemma 1, H 2 w is embedded in the Bergman-type space A i,δ C 1/2 , the exponent δ = δ(w) being defined in (2.11).Bounded Carleson measures for both spaces H 2 w and A i,δ C 1/2 have been compared in [25,26,9].We extend their results.Lemma 6.Let µ be a positive Borel measure on C 1/2 . ( From the estimate of Zw (2.7) and Lemma 5, µ is a Carleson measure for .
As for vanishing Carleson measures, the reasoning used in [9] for B 2 β can be transfered to the spaces A 2 β , with the test functions , We also require an equivalent norm for A i,δ C 1/2 , when δ > 0. For Bergman spaces of the unit disk, recall the following consequence of Stanton's formula [32,31]: Via the mapping τ 1/2 , we obtain that, for any f holomorphic on 2

BOUNDEDNESS OF Tg
In this section, we characterize functions in Xw, and prove Theorem 1.

4.1.
Carleson measure characterization.The boundedness of Tg on H 2 w can be described in terms of Carleson measures.This generalizes the setting of the Hardy space H 2 [14].
Recall that H 2 w is associated to the probability measure µw on the polydisk D ∞ .Proposition 3. Tg is bounded on H 2 w if and only if there exists a constant C = C(g) such that or, equivalently (4.17) The smallest constant C satisfying (4.16) is such that C ≍ Tg L(H 2 w ) .

Proof of Theorem 1 (a):
Bg depends on a finite number of variables.For 1 ≤ q and d ≥ 1, recall that f ∈ H q d,w if and only if f is in H q w and Bf is a function of z1, where R is the operator We define the set , for ǫ > 0.
For the moment, we admit that Jα ≤ C(d, w) , which will be proved in Lemma 7. Hence, Combining Lemma 3 with the following observation, Recall that Proof.When p −2 < T < 1, the change of variables u = T p 2x gives Next suppose that 0 < T ≤ p −2 .Since (log T ) 2 ≥ 4(log p) 2 , we notice that Proof of Lemma 7. Resorting to polar coordinates, and using changes of variables, we have We observe that Q = (x, t) ∈ (0, 1) In addition, since 0 √ t l ≤ 1, and we see that . Thus

It follows that Jα
We will obtain the Lemma by showing that (4.20) When β ≥ 1, we use that, for (x, t) ∈ Qk , and l = k, M p 2x l t l ≤ M (t l ), altogether with Lemma 8. We derive (4.20) from Hence, we see that J α,k J1 + J2,, where, by Lemma 8 and the relation p A change of variables provides the desired estimate.

Proof of Theorem 1 (b) and (c).
If f (s) = +∞ n=1 ann −s and g(s) = +∞ n=1 bnn −s , we have As in the case of H 2 , the operator is compact on Hw.Thus, set b1 = 1, and our study will be unchanged if we replace Tg by then g is in Xw, and the operator norms satisfy . Since w k ≤ w kl for any integers k, l, the Lemma is proven by the inequality We will also use the sufficient condition proved in Theorem 2.3 in [14], stating that if g is in BM OA(C0) ∩ D, then Tg is bounded on H 2 , with (4.21) Tg H 2 g BM OA(C 0 ) .

Proof of Theorem 1 (b) and (c).
If g is in BM OA(C0), Tg is bounded on H 2 , and (b) is a consequence of (4.21) and Lemma 9.
To prove (c), we use that (Tgf ) ′ = f g ′ , and that H 2 w is embedded in A i,δ C 1/2 , with δ = δ(w) > 0. We set Now formula (3.15), the boundedness of Tg on H 2 w and Lemma 1 induce that ) .We conclude by the characterization of the Bloch space given in Lemma 4.
We get a result which is in agreement with the situation for Hardy spaces [16], Bergman spaces [2] or the Hardy space of Dirichlet series H 2 [14], with the same proof.
Corollary 1.If g is in Xw, then g is in ∩0<p<∞H p w , and there exists c > 0, such that the function e c|Bg| is integrable on D ∞ , with respect to dµw.

COMPACTNESS
We now present a little oh version of Theorem 1.If the symbol is a vector of the standard orthonormal basis of H 2 w , that is we approximate Tg in the operator norm by the compact operator TS N g .Therefore, Tg is compact (see [14]).
The little oh version of Theorem 1 is related to the properties of V M OA(C0) ∩ D, and with the concept of vanishing Carleson measures.Theorem 3. Let g be in D. ( Proof.In order to prove (1), we use that V M OA(C0) ∩ D is the closure of Dirichlet polynomials in BM OA(C0) (see [14]), and that, from Theorem 1, we have Tg L(H 2 w ) g BM OA(C 0 ) .
Recall that H 2 w is embedded in A i,δ (C 1/2 ), δ = δ(w) being defined in (2.11).Assume that Tg is compact on H 2 w , and consider the measure Let (f k ) k be a weakly compact sequence in H 2 w .Formula (3.15), and Lemma 1 imply that By the compactness of Tg, νg is a vanishing Carleson measure for A i,δ (C 1/2 ), with , by the characterization of vanishing Carleson measures (Lemma 5).

MEMBERSHIP IN SCHATTEN CLASSES
Let g be a non constant symbol.As in the case of H 2 , the Volterra operator Tg on H 2 w does not belong to any Schatten class.Theorem 4. If the Dirichlet series g(s) = n≥2 bnn −s is not 0, then Tg : H 2 w → H 2 w is not in the Schatten class Sp, for any 0 < p < ∞.
Proof.Recall that (ew,n)n is an orthonormal basis of H 2 w .We follow the reasoning of Theorem 7.2 [14].Using that wNn wN wn, we see that, for N ≥ n, Tgew,n 2 For p ≥ 2, we obtain Therefore Tg is not in Sp for p ≥ 2, neither for 0 < p < ∞.

EXAMPLES
In this section, we study the boundedness of Tg on H 2 w , for specific symbols g.We consider fractional primitives of translates of the weighted Zeta function Zw and homogeneous symbols, which are the counterparts of the symbols presented in [14] in the H 2 setting.The techniques of proof, as well as the results are similar to theirs, and we omit the details.

Fractional primitives of translates of Zw.
Proposition 4. With the notation of (2.11) Proof.Abel summation and the Chebyshev estimate induce that g is in We adapt the test functions of [14], and take fJ (s . Now, for J a non-empty subset of {1, • • • , J}, we set nJ = j∈J pj , and observe that First assume that γ ≥ 0. From the prime number Theorem, we obtain that Therefore, it follows again from the prime number Theorem that for some constant c > 0, and Tg is unbounded.The case when γ < 0 is similar.
7.2.Homogeneous symbols.An m-homogeneous Dirichlet series has the form We extend Theorem 4.2 in [14] to the spaces H 2 w .
Proposition 5.There exist weights Wm(n) such that for g(s) = Ω(n)=m bnn −s , Precisely, there exist absolute constants Cm for which Moreover, when m = 2, log 2 n cannot be replaced in (7.22) by (log 2 n) 1+ε for any ε > 0.
Proof.If a linear symbol (m = 1) g(s) = p∈P bpp −s belongs to H 2 , we observe that g 2 . Hence, it follows from Theorem 4.1 in [14] and Lemma 9 that Tg is bounded on H 2 w and Tg L(H 2 w ) ≤ Tg L(H 2 ) .One can choose C1 = max (β + 1) −1 , 2 −β .(7.22) is a consequence of Theorem 4.2 in [14] and Lemma 9. We now prove the sharpness of the factor log 2 n.We assume that for some ε > 0, every 2-homogeneous Dirichlet series g satisfies (7.23) For x a large real number, and q ∼ e x a prime number, the symbol considered in [14] is We take as test functions If Sx denotes the set of square-free integers generated by the primes x/2 < p ≤ x, we have fx 2 , where N (x) := π(x) − π(x/2).Now, If n ∈ Sx, and p|n, we have , and wnq = wnwq.Thus, x) , and (7.23) does not hold, due to We will exhibit an homogeneous symbol g which is in H 2 w ∩ Bloch0(C 1/2 ), but not in Xw.In fact, we observe that g is in every H p w .
Lemma 10.If g is an m-homogeneous Dirichlet series in H 2 w , then g is in ∩0<p<∞H p w and, for any 0 < p < ∞, there exists c = c(m, p) such that Proof.It is enough to consider the case p ≥ 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 [30].For any integer m, there exists a constant C(m), such that max (wn, Now, suppose that, for some k, an m-homogeneous Dirichlet series h satisfies for any m.
We obtain that For general p, (7.24) is a consequence of H ölder's inequality.
For our construction, we need two technical Lemmas.
Under the condition 2η > 1, the sum S3(p1, p2) converges, and Integration by parts gives We handle the second integral via a change of variable:

Therefore
S3(p1, p2) (log L) 1−2η , L = log(p1p2).We next put M = log p1, and deal with With the notation f2(t) := (log 2 t) 2 [log (log t + M )] 2η−1 −1 , we obtain that where We derive The second integral is estimated in the same way: Therefore, we have Again, partial summation gives that when ε > , g ′ is convergent on C 1/2 , and its estimate near the line ℜs = 1 2 is determined by the behavior of the functions hj near the line ℜs = 1.Then g is in Bloch0(C 1/2 ), because of On another hand, the 3-homogeneous function  The coefficients in An satisfy bp 1 p 2 p 3 (log(p1p2p3)) log x x 3/2 (log 2 x) η+δ+1 .
and the proof is complete.

COMPARISON OF Xw WITH OTHER SPACES OF DIRICHLET SERIES
The previous results enable us to derive some inclusions involving Xw.In the context of the unit disk, the space of symbols g for which the Volterra operator Jg (1.3) is bounded on A 2 α (D) is Bloch(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 α (D).We shall study the counterparts of these facts for Xw.We say that Hg is bounded on The weak product H 2 w ⊙H 2 w is the Banach space defined as the closure of all finite sums Here the infimum is taken over all finite representations of F as We aim to relate Hankel forms and Volterra operators.The primitive of f ∈ D with constant term 0 is denoted by The Banach space ∂ −1 ∂H 2 w ⊙ H 2 w is the completion of the space of Dirichlet series F whose derivatives have a finite sum representation where the infimum is taken over all finite representations.The product rule (f g) It has been shown in [15] that, for the space As in the case of H 2 , the boundedness of Tg implies the boundedness of Hg.

Theorem 5. If the Volterra operator
Tg is bounded on H 2 w , then the Hankel form Hg is bounded.
Proof.We adapt the proof of Corollary 6.2 in [14] to the framework of the polydisk D ∞ .Polarizing the Littlewood-Paley formula (1), we get Then, we derive an expression of the half-Hankel form Since Tg is bounded on H 2 w , the Carleson measure characterization (4.16) induces that the form (8.29) is also bounded.Then Hg is bounded on H 2 w ⊙ H 2 w by the inclusion (8.28).
The previous Theorem states that we have The rest of the section is devoted to study the reverse inclusion.Let l 2 w denote the Hilbert space of complex sequences a = (an)n such that A sequence (ρn)n generates the following multiplicative Hankel form The symbol of the form is the Dirichlet series g(s) = n≥1 ρnn −s .The form ρ is said to be bounded if there is a constant C such that If f and h are Dirichlet series with coefficients a and b, respectively, we have When the symbol g has non negative coefficients, there is equivalence between the boundedness of Hg and the half-Hankel form (8.29).In fact, the proof given for H ).This differs from the case of weighted Dirichlet spaces on the unit disk, for which the boundedness of Hg, the form (8.29) and Tg 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 w is embedded in a Bergman space of the form A i,δ C 1/2 .For δ > 0, it is thus natural to define the Hankel form (8.31) Such multiplicative forms have been considered in the context of H 2 [13] and on A , where Proposition 8. Let δ > 0 as in (2.11).Then H (δ) defined in (8.31) is a multiplicative Hankel form with symbol φ δ , which is bounded on H 2 w ,0 ⊙ H 2 w ,0 .
Proof.The proof is similar to that of Theorem 13 in [10].The Cauchy-Schwarz inequality ensures that From Proposition 8 and the fact that the coefficients are positive, g is in H 2 w ,0 ⊗ H 2 w ,0 * for any 1 2 ≤ a < 1.In fact, the half Hankel form corresponding to g is bounded.We have seen in Proposition 4 that Tg is not bounded on H 2 w .Since Tg1 = g, g does not belong to X (H 2 w,0 ).In order to prove that g ∈ H 2  w ⊙ H In general, the dual of H 1 w is not known.However, it is shown in [10] that (8.32) where K is the space of Dirichlet series f (s) = +∞ n=1 ann −s such that The following consequence of this inclusion will stress upon the difference between the finite and infinite dimensional setting.
Proof.By Abel summation and the Chebyshev estimate, the symbol is in K, and thus in A 1 1 * .However, Tg is unbounded on A 2 1 (Proposition 4).
Proof.The inclusions have been proved in Theorem 1 and Corollary 1.As observed in [14], the symbols g(s) = Hence, they are not in BM OA(C0), though they belong to Xw (Lemma 9).The second inclusion is strict by Proposition 6.
With the method of Proposition 4, one can show that g(s) = n≥2 n −a log n n −s , 1/2 ≤ a < 1, is not in Xw, though it belongs to BM OA(C1−a) [14].Therefore, we have the strict inclusion Xw ⊂ = Bloch(C 1/2 ).Proof.The first inclusion has been shown in Theorem 1 (a).
If g is in D d ∩ Xw, Theorem 5 implies that Hg is bounded on H 2 w .Therefore, the form HBg (1.4) is bounded on the Bergman space H 2 w (D d ).From [18], Bg is in Bloch(D d ).Here is a function g which is not in Xw, such that Bg is in Bloch(D 2 ).Suppose that g ′ (s) = series are functions of the form (1.1) f (s) = +∞ n=1 ann −s , with s ∈ C.
On the finite polydisk D d (d ∈ N), the corresponding Bergman spaces H 2 w (D d ) -specifically B 2 β (D d ) and A 2 β (D d )are the L 2 −closures of polynomials with respect to the norm

Lemma 7 ..Lemma 8 .
There exists a constant C = C(w, d), such that The proof of Lemma 7 relies on technical computations (Lemma 8).For 0 < T < 1, and a real number p ≥ 2, set L := − log T 2 log p and K = min(1, L).There exists a constant C = C(p, w) > 0, such that d k=1 J α,k , where
Thus, νg is a Carleson measure for H 2 w and νg CM (H 2 w )

Case when Bg depends on a finite number of variables. We
Thus Tg is compact.It follows that every Dirichlet polynomial generates a compact Volterra operator on H 2 w .In general, if the symbol g(s) = n≥2 bnn −s satisfies an inequality of the form Tg 2 2tend to 0 as k → +∞.5.1.approximateasymbol g which is in Bloch0(C0) ∩ D d by a Dirichlet polynomial P in the Bloch(C0)-norm.From Theorem 1 (a), Tg is approximated in the operator norm by the compact operator TP .Theorem 2. If g is in Bloch0(C0) ∩ D d , then Tg is compact on H 2 w .5.2.Sufficient / necessary conditions for compactness. 1, = n |bn| 2 w −1 n ≍ n |bn| 2 ≍ S < ∞.By Lemma 10, g is in ∩0<p<∞H p w .It remains to prove that Tg is unbounded on H 2w .We again choose as test functions (cf the proof of Proposition 5) 8.1.Bounded Hankel forms.The Hilbert space H 2 w is equipped with the inner product ., .H 2 w .The Hankel form of symbol g ∈ D is defined on H 2 w by (8.27) Hg(f h) := f h, g H 2 w .
be a Banach space of Dirichlet series in which the space of Dirichlet polynomials P is dense.We say that a Dirichlet series φ is in the dual space Y * if the linear functional induced by φ via the H 2 w -pairing is bounded.In other words, φ ∈ Y * if and only if v φ (f ) = f, φ H 2 w , f ∈ P, extends to a bounded functional on Y. From its definition, Hg (8.27) is bounded on H 2 w if and only if g ∈ H 2 w ⊙ H 2 w * .

2 0
* is strict.As for the space H 2 w , the question whether the inclusion is strict remains open.The membership of g in ∂ −1 ∂H 2 w ⊙ H 2 Proposition 7 will enable us to provide examples of symbols g for which the Hankel form Hg and the half-Hankel form (8.29) are bounded, but the Volterra operator Tg is unbounded (see the proof of Proposition 9 [15]15]is valid for the spaces H 2 w .Proposition 7. Let g(s) = n≥1 ρnn −s be in H 2 w .The linear functional defined on H 2 w vg(f ) := f, g H 2 w is bounded on ∂ −1 ∂H 2 w ⊙ H 2 w if and only if the weighted form Jg(a, b) = g (∂ −1 (∂H 2 w ⊙H 2 w )) * ≍ vg ≍ |ρ1| + Jg .If ρ k ≥ 0 for all k, then g ∈ ∂ −1 ∂H 2 w ⊙ H2 w * if and only if g ∈ H 2 w ⊙ H 2 w * , with equivalent norms.
H 2 w + g (∞) f, g H 2 w , the first part of the proof entails that Hg is bounded on H 2 w ⊙ H 2 w .8.2.Xw and the dual of H 1 w .Keeping in mind the results known for Bergman spaces of the unit disk, it is natural to compare Xw and H 1 2w * , we consider the associated multiplicative form ρ (8.30).Let f, h be Dirichlet series with coefficients a, b, belonging to H 2 w .Sinceρ(a, b) = = Hg ((f − f (∞)) (g − g (∞))) + f (∞) h, g