Maxime Bailleul

Pascal Lefèvre

Pascal Lefèvre Some Banach

Some Banach spaces of Dirichlet series

published or not. The documents may come

Introduction

A Background and notations

In [START_REF] Hedenmalm | A Hilbert space of Dirichlet series and systems of dilated functions in L 2 (0, 1)[END_REF], the authors defined the Hardy space H 2 of Dirichlet series with square-summable coefficients. Thanks to the Cauchy-Schwarz inequality, it is a space of analytic functions on C 1 2 := {s ∈ C, (s) > 1 2 } and this domain is maximal. This space is isometrically isomorphic to the Hardy space H 2 (T ∞ ) (see [START_REF] Cole | Representing measures and Hardy spaces for the infinite polydisk algebra[END_REF] for the definition of H 2 (T ∞ )).

F. Bayart introduced in [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF] the more general class of Hardy spaces of Dirichlet series H p (1 ≤ p < +∞). We shall recall the definitions below.

In another direction, McCarthy defined in [START_REF] Mccarthy | Hilbert spaces of Dirichlet series and their multipliers[END_REF] some weighted Hilbert spaces of two types: Bergman-like spaces and Dirichlet-like spaces.

It is the starting point of many recent researches on spaces of Dirichlet series, for instance in [START_REF] Jf | Local properties of Hilbert spaces of Dirichlet series[END_REF], [START_REF] Olsen | On the boundary behaviour of the Hardy spaces of Dirichlet series and a frame bound estimate[END_REF] and [START_REF] Olsen | Local interpolation in Hilbert spaces of Dirichlet series[END_REF], some local properties of these spaces are studied and in [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF], [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF], [START_REF] Lefèvre | Essential norms of weighted composition operators on the space H ∞ of Dirichlet series[END_REF], [START_REF] Queffélec | Approximation numbers of composition operators on the H 2 space of Dirichlet series[END_REF] and [START_REF] Queffélec | Decay rates for approximation numbers of composition operators[END_REF] some results about composition operators on these spaces are obtained.

We recall some known facts about Dirichlet series. The study of Dirichlet series may appear more complicated than the study of power series. For instance, there is a first important difference: all the notions of radius of convergence coincide for Taylor series but Dirichlet series has several abscissas of convergence. The two most standard ones are the abscissa of simple convergence σ c and the abscissa of absolute convergence σ a (see [START_REF] Queffélec | Diophantine Approximation and Dirichlet Series[END_REF], [START_REF] Tenenbaum | Introduction a la théorie analytique et probabiliste des nombres. Cours spécialisés[END_REF]).

Let f be a Dirichlet series of the following form

f (s) = +∞ n=1
a n n -s (1).

We shall need the two other following abscissas: Actually, the two previous abscissas coincide: for all Dirichlet series f , one has σ b (f ) = σ u (f ) (see [START_REF] Bohr | Lösung des absoluten konvergenzproblems einer allgemeinen klasse dirichletscher reihen[END_REF]). This result due to Bohr is really important for the study of H ∞ , the algebra of bounded Dirichlet series on the right half-plane C + (see [START_REF] Maurizi | Some remarks on the algebra of bounded Dirichlet series[END_REF]). We shall denote by • ∞ the norm on this space:

σ u (f ) = inf{a | The series (
f ∞ := sup (s)>0 |f (s)|.
Let us recall now the principle of Bohr's point of view on Dirichlet series: let n ≥ 2 be an integer, it can be written (uniquely) as a product of prime numbers n = p α1 1 • • p α k k where p 1 = 2, p 2 = 3 etc . . . If s is a complex number and if we consider z = (p -s 1 , p -s 2 , . . . ), then we have by (1)

f (s) = +∞ n=1 a n (p -s 1 ) α1 • • (p -s k ) α k = +∞ n=1 a n z α1 1 • • z α k k .
So we can see a Dirichlet series as a Fourier series on the infinite-dimensional polytorus T ∞ . We shall denote this Fourier series D(f ). This correspondence is not just formal. For instance, let P be the set of prime numbers, Bohr proved the next result.

Theorem ( [START_REF] Bohr | Lösung des absoluten konvergenzproblems einer allgemeinen klasse dirichletscher reihen[END_REF]). Let f be a Dirichlet series of the form [START_REF] Aleman | Fourier multipliers for Hardy spaces of Dirichlet series[END_REF].

Then p∈P |a p | ≤ f ∞ .
The infinite-dimensional polytorus T ∞ can be identified with the group of complex-valued characters χ on the positive integers which satisfy the following properties |χ(n)| = 1 ∀n ≥ 1, χ(nm) = χ(n) χ(m) ∀n, m ≥ 1.

To obtain this identification for χ = (χ 1 , χ 2 , . . . ) ∈ T ∞ , it suffices to define χ on the prime numbers by χ(p i ) = χ i and use multiplicativity. We shall denote by m the normalized Haar measure on T ∞ . Now, let us recall how one can define the Hardy spaces of Dirichlet series H p . We already precised the case p = ∞, nevertheless, it is easy to see that the following description also applies to the case p = ∞. We fix p ≥ 1. The space H p (T ∞ ) is the closure of the set of analytic polynomials with respect to the norm of L p (T ∞ , m). Let f be a Dirichlet polynomial, by the Bohr's point of view D(f ) is an analytic polynomial on T ∞ . By definition f H p := D(f ) H p (T ∞ ) . The space H p is defined by taking the closure of Dirichlet polynomials with respect to this norm. Consequently H p and H p (T ∞ ) are isometrically isomorphic. When p = 2, H 2 is just the space of Dirichlet series of the form (1) which verify

+∞ n=1 |a n | 2 < +∞.
Let D be the space of functions which admit representation by a convergent Dirichet series on some half-plane. When a function f belongs to D and σ > 0, we can define the function f σ ∈ D, the translate of f by σ, i.e. f σ (s) := f (σ + s). We can then define a map from D to D by T σ (f ) = f σ .

For θ ∈ R, C θ is the half-plane defined by {s ∈ C, (s) > θ}.

We shall denote by P the space of Dirichlet polynomials, that is to say the vector space spanned by the functions e n (z) = n -z , where n ≥ 1. At last, in the sequel, for p ≥ 1, we write p its conjugate exponent:

1 p + 1 p = 1.

B Organization of the paper

In the present paper, we introduce two classes of Bergman spaces of Dirichlet series. We give some properties of these spaces, precise the growth of the point evaluation of functions belonging to these spaces. At last, we compare them to the Hardy spaces of Dirichlet series: it appears some very different phenomena. Definition 1. Let p ≥ 1, P be a Dirichlet polynomial and µ be a probability measure on (0, +∞) such that 0 ∈ Supp(µ). We define

P A p µ = +∞ 0 P σ p H p dµ(σ) 1/p .
A p µ will be the completion of P with respect to this norm. When µ(σ) = 2e -2σ dσ, we denote these spaces simply A p . More generally, let us fix α > -1 and consider the probability measure µ α , defined by

dµ α (σ) = 2 α+1 Γ(α + 1)
σ α exp(-2σ) dσ.

The space A p µα will be denoted simply A p α in this case.

Definition 2. On the infinite dimensional polydisk D ∞ , we consider the measure A = λ ⊗ λ ⊗ • • • where λ is the normalized Lebesgue measure on D. For p ≥ 1, the space B p (D ∞ ) is the closure of the set of analytic polynomials with respect to the norm of L p (D ∞ , A). Let f be a Dirichlet polynomial, we set

f B p := D(f ) B p (D ∞ ) .
The space B p is defined by taking the closure of P with respect to this norm.

In section 2, we prove that the point evaluation is bounded on the spaces A p µ for any s ∈ C 1/2 . More precisely, let δ s be the operator of point evaluation at s ∈ C 1/2 , which is a priori defined for Dirichlet polynomials (or convergent Dirichlet series). We prove that the operator extends to a bounded operator which we still denote by δ s and we obtain that there exists a constant c p such that for every s

∈ C 1/2 , δ s (A p ) * ≤ c p (s) 2 (s) -1 2/p .
We also show that the identity from H 2 to A p is not bounded when p > 2 but is compact when p = 2. Finally we obtain a Littlewood-Paley formula for the Hilbert spaces A 2 µ .

In section 3, we prove that the point evaluation is bounded on the space B p for any s ∈ C 1/2 and we have

δ s (B p ) * = ζ(2 (s)) 2/p .
By a result of hypercontractivity, we obtain that the injection from H p to B 2p is bounded. This phenomenon is similar to what happens in the classical framework of Hardy/Bergman spaces in one variable. Nevertheless, concerning compactness, we have the following curiosity: the injection from H p to B p is not compact. We also obtain a Littlewood-Paley formula for the space B 2 .

2 The Bergman spaces A p µ A Hilbert spaces of Dirichlet series with weighted 2 norm First, we recall some facts of [START_REF] Mccarthy | Hilbert spaces of Dirichlet series and their multipliers[END_REF]. We changed the definition in order to include the constants in these spaces, which seems to us more natural.

Let w = (w n ) n≥1 be a sequence of positive numbers, the space A 2 w is defined by

A 2 w := f ∈ D, f (s) = +∞ n=1 a n n -s , +∞ n=1 |a n | 2 w n < +∞ . Of course, if w ≡ 1, A 2
w is just the classical Hardy space H 2 . In order to obtain good properties for these spaces, we need to impose some properties on the weights.

Definition. [START_REF] Mccarthy | Hilbert spaces of Dirichlet series and their multipliers[END_REF] Let µ be a probability measure on (0, +∞) such that 0 ∈ Supp(µ). We define for n ≥ 1

w n := +∞ 0 n -2σ dµ(σ).
In this case, we say that the space A 2 µ := A 2 w is a (hilbertian) Bergman-like space and that w is a Bergman weight.

Example. When µ = δ 0 , the Dirac mass at point 0, we get the Hardy space H 2 . In the opposite situation, when µ({0}) = 0, it is easy to see that the sequence w converges to 0.

In the case µ = µ α , where α > -1, we have w n = (log(n) + 1) -1-α for n ≥ 1 and the associated space is A 2 α . For α = 0, we recover the space A 2 and we can notice that the limit (degenerated) case α = -1 corresponds to H 2 . Mac Carthy proved that these spaces are spaces of analytic functions on C 1 2 . It is a consequence of the following lemma:

Lemma ( [START_REF] Mccarthy | Hilbert spaces of Dirichlet series and their multipliers[END_REF]). Let w be a Bergman weight. Then w is non-increasing and w decreases more slowly than any negative power of n, that is to say:

∀ε > 0, ∃c > 0, w n > c n -ε ∀n ≥ 1.
In addition, C 1 2 is a maximal domain. Indeed let us consider the Riemann Zeta function ζ ( [START_REF] Tenenbaum | Introduction a la théorie analytique et probabiliste des nombres. Cours spécialisés[END_REF]), for every ε > 0 and every weight w:

ζ 1 2 + s + ε = +∞ n=1 1 n 1/2+ε+s ∈ A 2 w .
But these Dirichlet series admit a pole at 1 2 -ε.

B Point evaluation on A p µ

First we can easily compute the norm of the evaluation in the case of the Hilbert spaces A 2 µ . In this case the point evaluation is bounded on C 1/2 , it is optimal and the reproducing kernel

at s ∈ C 1/2 is K µ (s, w) := +∞ n=1 n -w-s w n and δ s (A 2 µ ) * ≤ +∞ n=1 n -2σ w n 1/2 for every s = σ + it ∈ C 1/2 .
In the general case, the next theorem provides us a majorization which gives the right order of growth when the abscissa is close to the critical value 1/2. Actually we are going to distinguish the behavior according to the valuation of the function, so we shall need some estimates according the constant coefficient vanishes or not. It would be interesting to work with truncated functions with higher order i.e. when each a n = 0 for n ≤ v, nevertheless we shall only concentrate on the case v = 0 and v = 1, because these are the only needed cases in this paper. Definition 3. Let H p ∞ be the subspace of H p of functions whose valuation is at least 1, i.e. the space of Dirichlet series whose constant coefficient a 1 vanishes (remember that a 1 is actually the value at infinity, and this explains our notation).

Let A p µ,∞ be the subspace of A p µ of functions whose constant coefficient vanishes. In the particular case of the measure µ α , we write A p α,∞ . At last, when α = 0, we simply use the natural notation A p ∞ . On the spaces H p (resp. H p ∞ ), we define ∆ p (s) (resp. ∆ p,∞ (s)) as the norm of the evaluation at point s ∈ C 1/2 . We recall that we know from [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF] that

∆ p (s) = ζ(2 (s)) 1/p .
Theorem 1. Let p ≥ 1 and µ be a probability measure on (0, +∞) such that 0 ∈ Supp(µ).

Then the point evaluation is bounded on P ∩ A p µ (resp. on P ∩ A p µ,∞ ) for any s ∈ C 1/2 . Hence it extends to a bounded operator on A p µ (resp. on A p µ,∞ ) whose norm verifies

(i) δ s (A p µ ) * ≤ inf η∈(0, (s)-1/2) ∆ p ( (s) -•) L p ([0, (s)-1/2-η], dµ) µ([0, (s) -1/2 -η]) . (ii) δ s (A p µ,∞ ) * ≤ inf η∈(0, (s)-1/2) ∆ p,∞ ( (s) -•) L p ([0, (s)-1/2-η], dµ) µ([0, (s) -1/2 -η]) .
Proof. We prove only (i) since the proof for (ii) is the same. Let us fix η in (0, (s) -1/2). We can assume that s = σ ∈ (1/2, +∞) thanks to the vertical translation invariance of the norm on A p µ . Let P be a Dirichlet polynomial. We have P (σ) = P ε (σ -ε) for any ε ∈ (0, σ -1/2).

We know that the point evaluation is bounded on H p :

|P (σ)| ≤ ∆ p (σ -ε) P ε H p for any ε ∈ (0, σ -1/2).
By integration on (0, σ -1/2 -η) we obtain

µ([0, σ -1/2 -η])|P (σ)| ≤ σ-1/2-η 0 ∆ p (σ -ε) P ε H p dµ(ε) .
Then, by Hölder's inequality,

µ([0, σ -1/2 -η])|P (σ)| ≤ P A p µ • ∆ p (σ -• ) L p ([0, σ-1/2-η],dµ) .
Since η ∈ (0, (s) -1/2) is arbitrary, the result follows.

Corollary 1. Let p ≥ 1 and α > -1.

(i) The point evaluation is bounded on A p α for any s ∈ C 1/2 and there exists a positive constant c p,α such that for every s ∈ C 1/2 we have:

δ s (A p α ) * ≤ c p,α (s) 2 (s) -1) (2+α)/p • (ii)
The point evaluation is bounded on A p α,∞ for any s ∈ C 1/2 and there exists a positive constant c p,α such that for every s ∈ C 1/2 we have:

δ s (A p α,∞ ) * ≤ c p,α (2 (s) -1) (2+α)/p • Proof.
In this proof, we shall use that, for every x > 1:

ζ(x) ≤ x x -1
• On the other hand, in the sequel, A B means that there exists some constant c depending on p and α only such that A ≤ cB.

Fix s = σ ∈ (1/2, +∞) and η ∈ (0, σ -1/2). In our framework, there exists some constant C α depending on α only, such that, for every A > 0:

µ α ([0, A]) ≥ C α min 1, A α+1 .

Let us prove (i).

We first consider the case p = 1. We choose η = (σ -1/2)/2. Since

sup ε∈[0, (σ-1/2)/2] |ζ(2σ -2 ε)| = ζ(σ + 1/2) ≤ 2σ + 1 2σ -1
the conclusion follows from the preceding theorem. Now let us assume that p > 1 and p = 2 (we already know exactly the norm of the evaluation in this case). We have

σ-1/2-η 0 ζ(2σ -2ε) p /p dµ α (ε) (2σ) p /p σ-1/2-η 0 ε α (2σ -2ε -1) p /p e -2ε dε.
We split our discussion in two cases, according to p > 2 or 2 > p > 1. First let us assume that p > 2. We have p /p < 1 hence the previous integral converges for η = 0 and is majorized by

σ-1/2 0 ε α (2σ -2ε -1) p /p dε = 1 (2σ -1) p /p • σ-1/2 0 ε α 1 -(2ε)/(2σ -1) p /p dε = (2σ -1) α+1 2 α+1 (2σ -1) p /p • 1 0 t α (1 -t) -p /p dt with t = 2ε 2σ -1 •
Finally we obtain:

σ-1/2-η 0 ζ(2σ -2ε) p /p dµ α (ε) (2σ) p /p B(α + 1, 1 -p /p) (2σ -1) (p /p)-α-1
where B is the classical Beta function ( [START_REF] Davis | Gamma function and related functions[END_REF]). Finally with the choice η = 0 in Theorem 1, we obtain

δ σ (A p α ) * B(α + 1, 1 -p /p) (2σ -1) (p /p)-α-1 1/p • (2σ) 1/p min 1, (σ -1/2) α+1 .
This estimation is good when σ is bounded (and more precisely when σ is close to 1/2). We have to look at the asymptotic behavior. So, coming back to the integral and considering σ ≥ 1, we have

σ-1/2 0 ζ(2σ -2ε) p /p dµ α (ε) we majorize it by σ-1 0 sup x≥2 |ζ(x)| p /p dµ α (ε) + σ-1/2 σ-1 ζ(2σ -2ε) p /p dµ α (ε).
The first integral is uniformly bounded relatively to σ and the second one is majorized by

σ-1/2 σ-1 ε α (2σ) p /p (2σ -2ε -1) p /p e -2ε dε σ α+p /p e -2σ 1 0 1 u p /p du 1.
It proves that the norm of the evaluation is uniformly bounded when σ > 1.

Gathering everything, the conclusion follows and (i) is proved when p > 2.

Now for the case 1 < p < 2, we have p /p > 1 and we cannot choose η = 0 because the integral is not convergent. But in fact, it suffices to choose the middle point η = (σ -1/2)/2. We conclude in the same way.

Let us prove (ii). Obviously δ s (A p α,∞ ) * ≤ δ s (A p α ) * , hence the conclusion follows from (i) when the real part of s is bounded by 1.

It suffices to look at the behavior when σ > 1 and it will follow from the (asymptotic) behavior of ∆ p,∞ :

∆ p,∞ (s) ≤ 1 (s) -1 .
Indeed, for every

f ∈ P ∩ H p ∞ ⊂ H 1 ∞ , we have for any s ∈ C 1 : f (s) = lim T →+∞ T -T ζ(s + it)f (it).
where

ζ(z) = n≥2 n -z . Hence |f (s)| ≤ ζ σ H ∞ f H 1 ≤ 1 σ -1 f H p .
Now, the sequel of the proof follows the lines of the proof of (i) so we leave the details to the reader.

Remarks.

(i) Let us precise why it is optimal in many cases: the behavior of δ s (A p α ) * around the critical line σ = 1/2 cannot be a power of (s) 2 (s) -1 better than (2 + α)/p. Indeed, let σ > 1/2, we would like to consider the function (ζ σ ) 2/p . Let us mention that we can define the function ζ q (where q > 0) through the Euler product:

ζ q (z) = p∈P 1 1 -p -z q .
Actually we first work with F being a partial sum of (ζ σ ) 2/p , we obtain:

|F (σ)| p ≤ δ σ p (A p α ) * F p A p α δ σ p (A p α ) * +∞ 0 (F ε ) p H p ε α exp(-2ε) dε
because F is a Dirichlet polynomial. Now if we assume that p > 1, we know (see [START_REF] Aleman | Fourier multipliers for Hardy spaces of Dirichlet series[END_REF]) that (e n ) n≥1 is a Schauder basis for H p hence there exists c p > 0 such that:

|F (σ)| p c p δ σ p (A p α ) * +∞ 0 ζ 2/p σ+ε p H p ε α exp(-2ε) dε. But (ζ σ+ε ) 2/p p H p = ζ σ+ε 2 H 2 = ζ(2σ + 2ε
) and we get (since F was an arbitrary partial sum of

ζ 2/p σ ): |ζ(2σ)| 2 δ σ p (A p α ) * n≥1 n -2σ (1 + ln(n)) α+1 . When α < 0, we get |ζ(2σ)| 2 δ σ p (A p α ) * (2σ -1) α hence 1 (2σ -1) (2+α) δ σ p (A p α ) *
which proves our claim, in a strong way: the majorization in (i) of Cor.1 is actually also (up to a constant) a minoration.

When α ≥ 0, we have

1 (2σ -1) (2+α) | log(2σ -1)| δ σ p (A p α ) *
which proves that we cannot get a better exponent than

(2 + α)/p in (i), Cor .1.
(ii) Let σ > 1/2 and µ = µ α , we already know that the reproducing kernel at σ is defined by

K µα (σ, w) = +∞ n=1 (1 + log(n)) α+1 n -σ-w , ∀w ∈ C 1/2 . Then K µα (σ, σ) ≤ δ σ (A 2 α ) * K µα (σ, • ) A 2 α
and by the property of the reproducing kernel

K µα (σ, σ) 1/2 ≤ δ σ (A 2 α ) * .
The converse inequality is already known, then

δ σ (A 2 α ) * = K µα (σ, σ) 1/2 = Γ(2 + α) (2σ -1) 2+α + O(1) 1/2
when σ goes to 1/2 (see [START_REF] Olsen | Local interpolation in Hilbert spaces of Dirichlet series[END_REF] for the second equality) and so our result is sharp when p = 2.

(iii) With the same notations, we have

K µα (σ, σ) 2 ≤ δ σ (A 1 α ) * K µα (σ, • ) 2 A 1 α = δ σ (A 1 α ) * K µα (σ, • ) 2 A 2 α
and again by the property of the reproducing kernel, we obtain

K µα (σ, σ) ≤ δ σ (A 1 α ) * .
We conclude as in (ii) and so the result is also sharp for p = 1.

(iv) In (i), we used that (e n ) n≥1 is a Schauder basis for H p when p > 1.

This result is also true for A p µ when p > 1: just use the result on H p , then it suffices to make an integration and use the density of the Dirichlet polynomials. This remark is also true for the spaces B p .

Let us mention here that we are able to give a more precise majorization in the particular case of an even integer p: constants are equal to 1. It immediately follows from a general method explained in annexe at the end of our paper: Proposition 1. Let p be an even integer, µ be as in Th.1.

(i) For every s ∈ C 1/2 we have:

δ s (A p µ ) * ≤ δ s 2/p (A 2 µ ) * .
In particular, (ii) For every s ∈ C 1/2 we have:

δ s (A p ) * ≤ (ζ -ζ )(2 (s)) 1/p ∼ 1 (2 (s) -1) 2/p when (s) → 1/2.
As soon as a Bergman like space is defined, a Dirichlet like space is naturally associated: Definition 4. Let p ≥ 1 and µ be a probability measure on (0, +∞). We define the Dirichlet space D p µ as the space of Dirichlet series f such that

f p D p µ := |f (+∞)| p + f p A p µ < +∞. Here f (+∞) stands for lim (s)→+∞ f (s) = a 1
, where f has an expansion (1).

Theorem 2. Let p ≥ 1 and µ be a probability measure on (0, +∞). For any s ∈ C 1/2 , we have

|f (s)| ≤ 2 1/p max 1, +∞ (s) δ t (A p µ,∞ ) * dt × f D p µ .
Proof. Without loss of generality we may assume that s = σ ∈ (1/2, +∞). Now

|f (σ)-f (+∞)| = +∞ σ f (t)dt ≤ +∞ σ |f (t)|dt ≤ +∞ σ δ t (A p µ,∞ ) * dt× f A p µ,∞
since the constant coefficient of f vanishes, i.e. f ∈ A p µ,∞ . So we get

|f (σ)| ≤ |f (+∞)| + +∞ σ δ t (A p µ,∞ ) * dt × f A p µ,∞ ≤ 1 + +∞ σ δ t (A p µ,∞ ) * dt p 1/p × |f (+∞)| p + f p A p µ,∞ 1/p
thanks to the Hölder's inequality. Now it suffices to remark that

1 + +∞ σ δ t (A p µ,∞ ) * dt p 1/p ≤ 2 1/p max 1, +∞ (s) δ t (A p µ,∞ ) * dt .
Corollary 2. Let α > -1 and p ≥ 1. There exists c p,α > 0 such that for every s ∈ C 1/2 , we have

δ s D p α ≤        c p,α 1 (2 (s) -1) (2+α) p -1 if α = p -2. c p,α log(2 (s) -1) if α = p -2.
Proof. The proof follows from Th. 2 and corollary 1.

Let us make a digression. The proofs of theorems 1 and 2 are based on the fact that we work with Bergman spaces with axial weights. Replacing axial weights by radial weights, the same idea can be adapted to classical Bergman and Dirichlet spaces on the unit disk D. Let us precise here how we can easily estimate the norm of the evaluation on weighted spaces of analytic functions over the unit disc.

Let σ : (0, 1) → (0, +∞) be a continuous function such that σ ∈ L 1 (0, 1). We extend it on D by σ(z) = σ(|z|). For p ≥ 1, we consider the weighted Bergman space

A p σ := H(D) ∩ L p (D, σ(|z|)dλ(z))
where H(D) is the set of analytic functions on D and λ is the normalized Lebesgue measure on D. This space is equipped with the norm

f A p σ = D |f (z)| p σ(z)dλ(z) 1/p .
We also consider the Dirichlet space D p σ (D): it is the space of analytic functions on D such that the derivative belongs to A p σ . This space is equipped with the norm

f D p σ = |f (0)| p + f p A p σ 1/p .
We know that the point evaluation at z ∈ D is bounded on the spaces H p (see [START_REF] Cowen | Composition operators on spaces of analytic functions[END_REF]) and we have

δ z H p = 1 (1 -|z| 2 ) 1/p .
δ z (A p σ ) * ≤ inf η∈(0,1-|z|) r → (1 -(|z|/r) 2 ) -1/p L p ([|z|+η, 1], σ(r)dr) S([|z| + η, 1])
where S(I) = I σ(r)dr.

Example. When σ ≡ 1 (The classical Bergman space A p ), we recover

δ z (A p ) * 1 (1 -|z| 2 ) 2/p
for any z ∈ D.

Theorem 4. Let p ≥ 1 and z ∈ D. We have

δ z (D p σ ) * ≤ 2 1/p max 1, |z| 0 δ r (A p σ ) * dr .
C A p µ is a space of Dirichlet series The results of the preceding section allow us to define, for each s ∈ C 1/2 , the value of f ∈ A p µ at s as δ s (f ). Of course, it coincides with the natural definition when f is a Dirichlet polynomial or when f ∈ D ∩ A p µ . Now we want more: we wish to check that we are actually working on spaces of Dirichlet series.

We first need the following tool.

Lemma 1. Let ε > 0 and µ be a probability measure on (0, +∞). Then

T ε P ∩ A 1 µ -→ A 2 µ f -→ f ε is bounded.
This extends to a bounded operator (still denoted T ε ) from A 1 µ to A 2 µ . In the proof, we shall use the following sequence.

Definition. Let µ be a probability measure on (0, +∞) with 0 ∈ Supp(µ). We define

w n := +∞ 0 n -σ dµ(σ) ∀n ≥ 1.
Proof. We shall introduce three bounded operators.

First we define S 1 :

P ∩ A 1 µ → H 1 by S 1 +∞ n=1 a n e n := +∞ n=1 a n w n e n .
S 1 is bounded because for any Dirichlet polynomial we have

N n=1 a n w n e n H 1 = T ∞ N n=1 a n w n z α1 1 • • z α k k dm(z) = T ∞ +∞ 0 N n=1 a n n -σ z α1 1 • • z α k k dµ(σ) dm(z) by definition of w n . ≤ +∞ 0 T ∞ N n=1 a n n -σ z α1 1 • • z α k k dm(z)dµ(σ) = N n=1 a n e n A 1 µ .
By density, this operator extends to a bounded operator (still denoted S 1 ). Now we define S 2 :

H 1 → H 2 by S 2 +∞ n=1 a n e n := +∞ n=1 a n n -ε √ w n e n .
S 2 is bounded because T ε/2 : H 1 → H 2 is bounded (see [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF]) and because, there exists C > 0 such that w n > Cn -ε .

The third operator S 3 :

H 2 → A 2 µ is defined by S 3 +∞ n=1 a n e n := +∞ n=1 a n √ w n e n .
S 3 is bounded because w n ≤ w n for all n ≥ 1.

Hence S 3 • S 2 • S 1 is bounded and clearly coincides with T ε .

Theorem 5. The space A p µ is a space of Dirichlet series: every

f ∈ A p µ ∈ D with σ u (f ) ≤ 1/2. Proof. It is obvious when p ≥ 2 since then A p µ ⊂ A 2 µ . When 1 ≤ p < 2, since A p µ ⊂ A 1
µ , we only have to prove the conclusion of the theorem in the case p = 1, but this follows from the preceding lemma. Indeed, let us fix

f ∈ A 1 µ , α > 1/2 and ε = α -1/2 > 0. The function T ε (f ) belongs to A 2 µ so we can write for every z ∈ C 1/2 T ε (f )(z) = n≥1 a (ε) n n -z .
On the other hand, f is a limit of a sequence of Dirichlet polynomials (P k ) k∈N relatively to the space A 1 µ . The continuity of T ε implies that T ε (f ) is the limit of P k (ε + •) k∈N relatively to the norm of A 2 µ . Invoking the continuity of the point evaluation both at z + ε and at z, we get

f (z + ε) = lim P k (ε + z) = lim T ε P k (z) = T ε (f )(z) = n≥1 a (ε) n n -z .
In particular, for every s ∈ C α , we have (with

z = s -ε ∈ C 1/2 ): f (s) = n≥1 a (ε) n n ε n -s .
Actually the coefficient do not depend on ε (by uniqueness of the Dirichlet expansion). Since α > 1/2 is arbitrary, we get the conclusion.

An immediate corollary of this section is the following proposition Proposition 2. In view of the results of this section, we actually have for every ε > 0 that

T ε A 1 µ -→ A 2 µ f -→ f ε is well defined and bounded. Recall that f ε (s) = f (s + ε) = δ s+ε (f ).
It seems clear that H p ⊂ A p µ for any p ≥ 1 and any µ. Indeed, the following theorem precise this fact and that the way we may compute the norm remains valid for general functions of A p µ .

Theorem 6. Let p ≥ 1 and µ a probability measure whose support contains 0.

(i) H p ⊂ A p µ and, for every f ∈ H p , we have

f A p µ ≤ f H p .
(ii) For every f ∈ H p , we have

f A p µ = +∞ 0 f σ p H p dµ 1/p . (iii) For every f ∈ A p µ , we have f A p µ = lim c→0 + f c A p µ .
Proof. For every Dirichlet polynomials f , we have f A p µ ≤ f H p , since µ is a probability measure and f H p = sup c>0 f c H p . Now, (in the spirit of the proof of Th.5) a density argument, combined with the boundedness of point evaluation, allows to conclude easily the first assertion.

Let f ∈ H p and ε > 0. There exists a Dirichlet polynomial P such that f -P H p < ε. By the first assertion f -P A p µ < ε and then

f A p µ ≤ ε + P A p µ = ε + +∞ 0 P σ p H p dµ(σ) 1/p ≤ ε + +∞ 0 P σ -f σ p H p dµ(σ) 1/p + +∞ 0 f σ p H p dµ(σ) 1/p
. Now, T σ is a contraction on H p for every σ > 0 and so

f A p µ ≤ 2ε + +∞ 0 f σ p H p dµ(σ) 1/p
.

By the same way we obtain a lower bound and finally the second assertion.

For the third assertion, we shall use that T c is a contraction on A p µ for every c > 0: as in the first assertion, it suffices to check it on Dirichlet polynomials but in this case this is clear by definition of the norm for Dirichlet polynomials and the fact that T c is a contraction on H p . Now let f ∈ A p µ and ε, c > 0. There exists P a Dirichlet Polynomial such that f -

P A p µ < ε, then f -f c A p µ ≤ f -P A p µ + P -P c A p µ + P c -f c A p µ ≤ 2 f -P A p µ + P -P c A p µ ≤ 2ε + P -P c A p µ
and by the dominated convergence theorem P -P c A p µ goes to 0 when c goes to 0 and so the result is proved.

D Vertical limits and Littlewood-Paley formula

Let f be a Dirichlet series absolutely convergent in a half-plane. For any sequences (τ n ) ⊂ R, we can consider vertical translations of f ,

(f τn (s)) := (f (s + iτ n )).
By Montel's theorem, this sequence is a normal family in the half-plane of absolute convergence of f and so there exists a convergent subsequence f . We say that f is a vertical limit of f . We shall use the next result.

Proposition ( [START_REF] Hedenmalm | A Hilbert space of Dirichlet series and systems of dilated functions in L 2 (0, 1)[END_REF]). Let f be a Dirichlet series of the form (1), absolutely convergent in a half-plane. The vertical limit functions of f are exactly the functions of the form

f χ (s) := +∞ n=1 a n χ(n) n -s where χ ∈ T ∞ .
In [START_REF] Hedenmalm | A Hilbert space of Dirichlet series and systems of dilated functions in L 2 (0, 1)[END_REF] it is shown that every element f in H 2 admits vertical limit functions f χ which converges m-almost everywhere on C + . We have the same result with the Bergman spaces A p µ . This is a consequence of the following lemma.

Menchoff 's Lemma ( [START_REF] Olevskiaei | Fourier series with respect to general orthogonal systems[END_REF]). Let (Ω, A, ν) be a probability space and (Φ n ) be an orthonormal sequence in L 2 (Ω).

Then +∞ n=1 |c n | 2 log 2 (n) < +∞ ⇒ +∞ n=1 c n Φ n converge ν -ae.
Proposition 3. Let p ≥ 1, µ be a probability measure on (0, +∞) and let f ∈ A p µ or D p µ . For almost all χ ∈ T ∞ , f χ converges on C + . Remark. It suffices to give the proof in the case of the Bergman-like spaces: indeed if f is in D p µ then f ∈ A p µ and so f χ converges on C + for almost every χ ∈ T ∞ and the same hold for f .

Proof. First, we prove the result when p = 2. Let f ∈ A 2 µ of the form (1) and c n := a n n -σ-it where σ > 0 and t ∈ R. Clearly (χ(n)) is an orthonormal family in L 2 (T ∞ ). We have:

+∞ n=2 |c n | 2 log 2 (n) = +∞ n=2 |a n | 2 w n log 2 (n) n 2σ w n .
If w is a Bergman weight, we know that there exists a positive constant C such that w n > Cn -σ for all n ≥ 1, so

log 2 (n) n 2σ w n ≤ log 2 (n)
Cn σ . In this case, the right term of the previous inequality is finite and by the Menchoff's lemma, the proof is finished for p = 2. Now we want to prove this result when p = 2. By inclusion on these spaces, it suffices to prove the result for p = 1.

Let f ∈ A 1 µ . By proposition 2, f ε ∈ A 2 µ for every ε > 0. So for every ε > 0, for almost all χ ∈ T ∞ , (f ε ) χ converges on C + .

Then we have: for every n ≥ 1, for almost all χ ∈ T ∞ , (f 1/n ) χ converges on C + .

Now we can invert the quantifiers:

for almost all χ ∈ T ∞ , ∀n ≥ 1, (f 1/n ) χ converges on C + .

Of course if (f 1/n ) χ converges on C + for every n ≥ 1, f χ converges on C + and so we obtain result. Now, following some ideas from [START_REF] Kellay | Compact composition operators on weighted Hilbert spaces of analytic functions[END_REF] in the case of the unit disk, we consider the case of the weighted Bergman-like spaces when dµ(σ) = h(σ)dσ where h ≥ 0, h L 1 (R + ) = 1 and 0 ∈ Supp(f ). Let w h be the associated Bergman weight defined for n ≥ 1 by

w h (n) = +∞ 0 n -2σ h(σ)dσ.
For σ > 0, we define

β h (σ) := σ 0 (σ -u)h(u) du = σ 0 t 0 h(u) dudt.
Remark. Point out that lim σ→+∞ β h (σ)n -2σ = 0 for every n ≥ 2.

We can compute the two first derivatives of β h :

β h (σ) = σ 0 h(u)du, β h (σ) = h(σ).
In order to obtain a Littlewood-Paley formula for the spaces A 2 µ , we need the following lemma.

Lemma ( [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF]). Let η be a Borel probability measure on R and f of the form (1).

Then f 2 H 2 = T ∞ R |f χ (it)| 2 dη(t)dm(χ).
Theorem 7.

["Littlewood-Paley formula"] Let η be a Borel probability measure on R. Then

f 2 Aw h = |f (+∞)| 2 + 4 T ∞ +∞ 0 R β h (σ)|f χ (σ + it)| 2 dη(t)dσdm(χ).
Proof. Let f ∈ A 2 w h of the form (1), the previous lemma applied to f σ where σ > 0 gives

T ∞ R |f χ (σ + it)| 2 dη(t)dm(χ) = +∞ n=2 |a n | 2 n -2σ log 2 (n) ∀σ > 0.

Now we multiply by β h (σ) and integrate over R

+ : +∞ 0 T ∞ R β h (σ)|f χ (σ+it)| 2 dη(t)dm(χ)dσ = +∞ n=2 |a n | 2 log 2 (n) +∞ 0 n -2σ β h (σ)dσ.

Now, it suffices to prove that

w h (n) = 4 log 2 (n) +∞ 0 n -2σ β h (σ)dσ.
But by definition, we have

w h (n) = +∞ 0 h(σ)n -2σ dσ.
An integration by parts gives

w h (n) = σ 0 h(u) du × n -2σ +∞ 0 + 2 log(n) +∞ 0 σ 0 h(u) du × n -2σ dσ = β h (σ) × n -2σ +∞ 0 + 2 log(n) +∞ 0 β h (σ) × n -2σ dσ.
But we know that β h (σ) → 0 when σ → 0 (because h ∈ L 1 (R + )) and β h (σ) → h 1 when σ → +∞ . So we have

w h (n) = 2 log(n) +∞ 0 β h (σ) × n -2σ dσ.
Using again an integration by parts, we obtain:

w h (n) = 4 log 2 (n) +∞ 0 n -2σ β h (σ)dσ.
Example. let α > -1 and f ∈ A2 α , we have β h (σ) ≈ σ α+2 when σ is small. Then

f 2 α ≈ |f (+∞)| 2 + 2 α+3 Γ(α + 3) T ∞ +∞ 0 R σ α+2 |f χ (σ + it)| 2 dη(t)dσdm(χ).
In the case of Dirichlet spaces, we obtain the following proposition.

Proposition 4. Let dµ = hdσ be a probability measure. Then for f ∈ D 2 µ we have

f 2 D 2 µ = |f (+∞)| 2 + 4 T ∞ +∞ 0 R h(σ)|f χ (σ + it)| 2 dη(t)dσdm(χ).
Remark. These formulas are really useful to prove some criterion for compactness of composition operators (see [START_REF] Bailleul | Composition operators on weighted Bergman spaces of Dirichlet series[END_REF]). We can also use these formulas to compare A 2 and H 2 norms. For example, assume that f ∈ A 2 ∞ . Then for x > 0, we have

f 2 A 2 ≥ 4 x 0 T ∞ R |f χ (σ + it)| 2 dη(t)dm(χ) σ 2 dσ.
But we know that

T ∞ R |f χ (σ + it)| 2 dη(t)dm(χ) = f σ 2 H 2 ≥ f x 2 H 2
when σ ≤ x. So we have

f 2 A 2 ≥ 4 f x 2 H 2 × x 0 σ 2 dσ and so f x H 2 ≤ √ 3 f A 2 2x 3 ∀x > 0.
Obviously we can do the same with the spaces A 2 µ . Corollary 3. Let ε > 0, we have:

T ε (A 2 µ ) ⊂ H 2 ⊂ A 2 µ .
E Comparison between A p and H p .

We already saw that H p ⊂ A p . The goal of this section is to prove the following theorem. Theorem 8. Let p > 2. The identity from H 2 to A p is not bounded but the identity from H 2 to A 2 is compact.

We need the following lemma (we did not find in the literature any such formula). Lemma 2. For n ≥ 1, we have:

+∞ k=0 n + k n 2 z k = 1 (1 -z) 2n+1 n k=0 n k Proof. Proof 1. For n = 1, we easily check that +∞ k=0 k + 1 1 2 z k = 1 + z (1 -z) 3 .
We can now prove the equality by induction just by noting that

n + k n × (n + k + 1) n + 1 = n + k + 1 n + 1 .
Now it suffices to compute the second derivative of the equality for the rank n to obtain the equality for the rank n + 1, nevertheless the computation is fastidious and we leave it to the reader.

Proof 2. We give also a quick and elementary argument. Fix z ∈ D. We

want to estimate S = ∞ k=0 n + k k 2 z k .
Since for every w ∈ D, we have

1 (1 -w) n+1 = ∞ k=0 n + k k w k , we point out that S = 1 n! G (n) (1) where G(w) = w n (1 -zw) n+1 •
Using now the Leibnitz formula, we get

S = 1 n! n k=0 n k n! k! • (n + k)!z k n!(1 -z) n+k+1 = 1 (1 -z) 2n+1 S where S = n k=0 n k n + k k z k (1 -z) n-k
, which is the derivative of order n at point w = z of the function

w -→ 1 n! n k=0 n k w n+k (1 -z) n-k = w n n! w + 1 -z n .
Hence with help of the Leibnitz formula once again, we obtain

S = 1 n! n k=0 n k n! k! z k • n! (n -k)! (z + 1 -z) n-k = n k=0 n k 2 z k .
We get the conclusion.

After this work was completed, M. De La Salle communicated to us a third proof which relies on the computation of a residue. Proposition 5. Let m ≥ 1 be an integer. There exists γ m > 0 such that

+∞ n=1 d m (n) 2 n -2σ ∼ γ m (2σ -1) m 2 when σ → 1/2.
Proof. We know that d m is multiplicative, so we have

+∞ n=1 d m (n) 2 n -2σ = p∈P k≥0 d m (p k ) 2 p -2σk .
Now, we can compute each series in the product because:

d m (p k ) = p α 1 .. p αm =p k α1,..., αm≥0 1 = α1+•••+αm=k α1,..., α k ≥0 1 = m + k -1 m -1 .
So, by the first lemma, we have

+∞ n=1 d m (n) 2 n -2σ = p∈P m-1 k=0 m-1 k 2 (p -2σ ) k (1 -(p -2σ ) k ) 2m-1 .

Now we have

m-1 k=0 m -1 k 2 z k × (1 -z) (m-1) 2 = Q(z)
where

Q(0) = 1 and Q (0) = 0 because the coefficient of z is m -1 1 2 - (m -1) 2 1 = (m -1) 2 -(m -1) 2 = 0.
We get that (Q(p -2σ )) p ∈ 1 when σ ≥ 1/2 and the infinite product p∈P Q(p -2σ ) is convergent and has a positive limit when σ → 1/2. Finally we obtain

+∞ n=1 d m (n) 2 n -2σ = p∈P Q(p -2σ ) p∈P (1 -p -2σ ) (m-1) 2 +2m-1 = p∈P Q(p -2σ ) p∈P (1 -p -2σ ) m 2 .
And so when σ → 1/2, we obtain

+∞ n=1 d m (n) 2 n -2σ = ζ(2σ) m 2 × p∈P Q(p -2σ ) ∼ γ m (2σ -1) m 2 .

An immediate corollary is

Corollary 4. Let m ≥ 1 be an integer, there exists c m > 0 such that

ζ m (σ + •) H 2 ∼ c m (2σ -1) m 2 /2
when σ → 1/2. Now we can prove Theorem 8.

Proof. of Th.8. Assume that the injection from H 2 to A p is bounded, then there exists m ≥ 1 such that

2 < 2(m + 1) m < p.
The identity from H 2 to A 2(m+1) m would be also bounded: there exists C > 0 such that, for every f ∈ H 2 ,

f A 2(m+1) m ≤ C f H 2 .
We apply this inequality to the m-th power of the reproducing kernels of

H 2 : s → ζ m (σ + s) with σ > 1/2. Then ζ m (σ + •) A 2(m+1) m ≤ C ζ m (σ + •) H 2
and thanks to the last corollary we know that

ζ m (σ + •) H 2 ∼ c m (2σ -1) m 2 /2 when σ → 1/2.
Now for the first term in the inequality we have

ζ m (σ + •) A 2(m+1) m = ζ m+1 (σ + •) m m+1 A 2 = +∞ n=1 d m+1 (n) 2 n -2σ log(n) + 1 m 2(m+1)
.

By the previous proposition, we know that

+∞ n=1 d m+1 (n) 2 n -2σ ∼ γ m+1 (2σ -1) (m+1) 2 when σ → 1/2.
So by integration, we obtain

+∞ n=1 d m+1 (n) 2 n -2σ log(n) + 1 m 2(m+1) ∼ γ m (2σ -1) m 2 (m+2) 2(m+1)
for some γ m > 0. Now using the inequality given by the boundedness of the identity, we obtain for σ close to 1/2

1 C × (2σ -1) m 2 (m+2) 2(m+1) -m 2 /2 = (2σ -1) m 2 2(m+1)
and this is obviously false.

To finish the proof we have to show that the injection from H 2 to A 2 is compact but in fact it suffices to remark that this injection is a diagonal operator on the orthonormal canonical basis e n n≥1 of H 2 : the eigenvalues, equal to 1 log(n) + 1

, tends to zero.

F Inequalities on coefficients

We shall give here some inequalities between the A p norm and some weighted p norms of the coefficient of the functions. This follows the spirit of classical estimates on Bergman spaces (see [START_REF] Duren | Bergman spaces[END_REF] p.81 for instance). Theorem 9. Let p ≥ 1 and µ be a probability measure on (0, +∞) such that 0 ∈ Supp(µ) and (w n ) n≥1 the associated weight.

(i) When 1 ≤ p ≤ 2 and f = n≥1
a n e n ∈ A p µ , we have

w 1/p n a n p ≤ f A p µ .
(ii) When p ≥ 2 and

n≥1 w p -1 n |a n | p < ∞, we have f = n≥1
a n e n ∈ A p µ and

f A p µ ≤ n≥1 w p -1 n |a n | p 1/p = w 1/p n a n p .
An immediate corollary is

Corollary 5. Let p ≥ 1. (i) When 1 ≤ p ≤ 2 and f = n≥1
a n e n ∈ A p , we have

a n (1 + ln(n)) 1/p p ≤ f A p . (ii) When p ≥ 2 and n≥1 |a n | p (1 + ln(n)) p -1 < ∞, we have f = n≥1 a n e n ∈ A p and f A p ≤ n≥1 |a n | p (1 + ln(n)) p -1 1/p = a n (1 + ln(n)) 1/p p .
Proof. of Th.9. Let us detail the case 1 ≤ p ≤ 2.

For every integer n

= p α1 1 p α2 2 . . . ≥ 1 and f ∈ L p (R + × T ∞ , dµ ⊗ dm), let us define τ n (f ) = R + ×T ∞ f (σ, z)z (n) n -σ dµ(σ) ⊗ dm(z)
where z (n) = z α1 1 z α2 2 . . . We point out that, when P is a Dirichlet polynomial P (s) = n≥1 a n n -s , we can associate as usual f (σ, z) = n≥1 a n n -σ z α1 1 z α2 2 . . . We have in that case

τ n (f ) = w n a n . Then we consider Q(f ) = τ n (f ) n≥1 . Q defines a norm one operator from L 1 (R + × T ∞ , dµ ⊗ dm) to L ∞ (ω) and from L 2 (R + × T ∞ , dµ ⊗ dm) to L 2 (ω)
, where L q (ω) is the Lebesgue space on positive integers with discrete measure whose mass at point n is given by 1/w n . Indeed:

|τ n (f )| ≤ f 1 and n≥1 |τ n (f )| 2 w n = n≥1 | f, b n | 2 where b n (σ, z) = z (n) n -σ / √ w n is an orthonormal system in the Hilbert space L 2 (R + × T ∞ , dµ ⊗ dm). So the Bessel inequality gives n≥1 |τ n (f )| 2 w n ≤ f 2 2 .
Now by interpolation (apply the Riesz-Thorin theorem), Q is bounded from

L p (R + × T ∞ , dµ ⊗ dm) to L p (ω): n≥1 |τ n (f )| p w n 1/p ≤ f p .
Writing this inequality in the particular case of f associated to a Dirichlet polynomial (as described as the beginning of the proof), the result follows.

The other case is obtained (it is even easier) in the same way.

3 The Bergman spaces B p

A The Bergman spaces of the infinite polydisk

Recall that A = λ ⊗ λ ⊗ • • • where λ is the normalized Lebesgue measure on the unit disk D. For p ≥ 1, B p (D ∞ ) is the Bergman space of the infinite polydisk. It is defined as the closure in the space L p (D ∞ , A) of the span of the analytic polynomials.

Remark. Let P be an analytic polynomial defined for z

= (z 1 , z 2 , . . . ) ∈ D ∞ by P (z) := N n=1 a n z α1 1 • • z α k k . Then P B 2 = N n=1 |a n | 2 (α 1 + 1) • • (α k + 1) 1/2 . So clearly H 2 (T ∞ ) ⊂ B 2 (D ∞ ).
In fact this is also true for any p ≥ 1, it suffices to apply several times this property in the case of the unit disk.

Recall that the Bergman kernel at z, w ∈ D is defined by k(w, z)

:= 1 (1 -wz) 2 .
Remark. Each element G n belongs to B 2 (D ∞ ) and we have

G n L 2 (D ∞ ) = G n B 2 (D ∞ ) ≤ G n H 2 (T ∞ ) = G n L 2 (T ∞ ) .
So {G n } is a bounded martingale in L 2 (D ∞ ). By the Doob's theorem and by closure we know that the product converges pointwise and in norm in B 2 (D ∞ ).

We need to recall some notations and results from [START_REF] Cole | Representing measures and Hardy spaces for the infinite polydisk algebra[END_REF].

Let U be a uniform algebra on a compact space X and µ be a measure on X. H p (µ) is the closure of U in L p (µ).

Proposition. ([7]

) Let U be a uniform algebra on a compact space X, µ be a probability measure on X, y ∈ X such that the evaluation extends continuously to H 2 (µ). Assume that any real power of the reproducing kernel of this point evaluation x → K(x, y) belongs to H 2 (µ). Then for p ≥ 1, we have

| f (y)| p ≤ K(y, y) |f (x)| p dµ(x)
for every function f in H p (µ) and the norm of the point evaluation at y is exactly K(y, y) 1/p . With the last remark and this theorem, we obtain that the point evaluation is bounded on

B p (D ∞ ) for ζ ∈ D ∞ ∩ 2 and we have |f (ζ)| p ≤ +∞ i=1 1 (1 -|ζ i | 2 ) 2 f p B p (D ∞ ) .
Moreover f is holomorphic on D ∞ ∩ 2 thanks to [START_REF] Cole | Representing measures and Hardy spaces for the infinite polydisk algebra[END_REF].

B Point evaluation on B p

In the sequel, R will denote the infinite product of the probability measures 2r i dr i on [0, 1]. Definition 7. Let P ∈ P of the form N n≥1 a n n -s . We define on P the norm

P B p := [0,1] ∞ N n=1 a n r α1 1 • • r α k k e n (it) p H p dR 1/p .
Remark. The fact that this defines a norm follows from the next proposition.

Definition 8. Let p ≥ 1. We denote by B p the closure of P relatively to the norm • B p : it is the Bergman space of Dirichlet series.

Remark. We denote d(n) the number of divisors of n. For f as in (1), one has

f B 2 = +∞ n=1 |a n | 2 d(n) 1/2 .
First we use the Bohr's point of view to precise the link between B p and B p (D ∞ ). (i) Let P ∈ P. We have

P B p = D(P ) B p . (ii) D : P → B p (D ∞ ) extends to an isometric isomorphism from B p onto B p (D ∞ ).
Proof. The first fact is clear. For the second one, remember that B p is the closure of P and that B p (D ∞ ) is the closure of the set of analytic polynomials.

Theorem 10. Let p ≥ 1 and f ∈ B p . The abscissa of uniform convergence of f verifies σ u (f ) ≤ 1 2 . Moreover, when (w) > 1 2 , we have

|f (w)| ≤ ζ(2Re(w)) 2/p f B p
and:

δ w = ζ(2Re(w)) 2/p .
In addition, there exists f ∈ B p such that σ b (f ) = 1/2.

Proof. Let f ∈ B p and s ∈ C 1 2 . We define

z s = (p -s 1 , p -s 2 , . . .) ∈ D ∞ ∩ 2 . We know that D(f ) ∈ B p (D ∞ ) and so |D(f )(z s )| p ≤ +∞ i=1 1 (1 -|p -s i | 2 ) 2 D(f ) p B p (D ∞ ) .
But thanks to the last proposition D(f

) B p (D ∞ ) = f B p and D(f )(z s ) = n=p α 1 1 ... p α k k n≥1 a n (p -s 1 ) α1 . . . (p -s k ) α k = n=p α 1 1 ... p α k k n≥1 a n (p α1 1 . . . p α k k ) -s = n≥1 a n n -s .
Then we have

|f (s)| p ≤ +∞ i=1 1 1 -p -2 (s) i 2 f B p = ζ(2 (s)) 2 f p B p .
So f admits a bounded extension on each smaller half-plane of C 1 2 . By the Bohr's theorem we have σ u (f ) ≤ 1 2 .

To prove that the norm of the evaluation is exactly ζ(2Re(w)) 2/p , it suffices to use the corresponding result from [START_REF] Cole | Representing measures and Hardy spaces for the infinite polydisk algebra[END_REF] on B p (D ∞ ).

C Comparison between B p and H p

In this section, we precise the link between B p and H p . This question is natural as soon as we keep in mind the behavior of the injection from H p (D) to B q (D) in the classical framework of one variable Hardy-Bergman spaces on the unit disk. We recall that H p ⊂ B q if and only if q ≤ 2p and that this injection is compact if and only if q < 2p (see [START_REF] Lefèvre | Finitely strictly singular operators in harmonic analysis and function theory[END_REF] for recent results on the limit case q = 2p).

First, following ideas of [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF], we obtain a result of hypercontractivity between the spaces B p .

Let 1 ≤ p ≤ q < +∞. For f ∈ B p (D ∞ ), z ∈ D ∞ and k ≥ 1, we define ẑk = (z 1 , . . . , z k-1 , z k+1 , . . . ). Let f ẑk (z k ) = f (z). Then D ∞ f ẑk r L r (D) dm( ẑk ) = f r L r (D ∞ ) .
We consider a sequence of operators S k : B p (D) → B q (D) for k ≥ 1 such that S k (1) = 1. If P is an analytic polynomial on D ∞ , we define

P 1 = P, P k+1 = S k (P ẑk (z k )).
This sequence is not in general a sequence of polynomials but if P depends of z 1 , . . . , z n then each term of this sequence too. So this sequence is stationary. Proposition 8. If +∞ k=1 S k < +∞ then the sequence (P k ) k≥1 converges to some S(P ) of B q (D ∞ ). In addition, S extends to a bounded operator from B p (D ∞ ) to B q (D ∞ ).

Actually, if we consider a sequence of operators S k : H p (D) → B q (D), we obtain the following similar result. Proposition 9. If +∞ k=1 S k < +∞ then the sequence (P k ) converges to some S(P ) of B q (D ∞ ). In addition, S extends to a bounded operator from H p (T ∞ ) to B q (D ∞ ).

We only give the proof of the second proposition.

Proof. It suffices to show that

P (k+1) B q (D ∞ ) ≤ k i=1 S i P H p (T ∞ ) .
One has

P (k+1) q B q (D ∞ ) = D ∞ |S k (P k ẑk (z k ))| q dA(z) = D ∞ D |S k (P k ẑk (z k ))| q dλ( ẑk )dA(z k ) = D ∞ S k (P k ẑk (.)) q B q (D) dA( ẑk ) ≤ S k q D ∞ P k ẑk (•) q H p (T) dA( ẑk ) = S k q D ∞ T |P k ẑk (χ k )| p dm(χ k )
q/p dA( ẑk ).

Since q p ≥ 1, we get, by the integral triangular inequality,

P (k+1) q B q (D ∞ ) ≤ S k q T D ∞ |P k ẑk (χ k )| q dm( ẑk ) p/q dm(χ k ) q/p
By induction, we obtain the result.

We shall give some applications of these propositions, but we first need other preliminaries, in the classical setting of the unit disk.In the following, for q ≥ 1, the space B q (D) (resp. H q (D)) is the classical Bergman space (resp. the classical Hardy space). Proof. We adapt a proof from [START_REF] Bonami | étude des coefficients de Fourier des fonctions de L p (G).(french)[END_REF]. Let f ∈ H 1 (D), with norm 1, of the form

f (z) = +∞ n=0 a n z n .
We considerer the factorisation f = gh where g and h are in H 2 (D) and verify |g| 2 = |h| 2 = 1. Denote (b n ) and (c n ) the Fourier coefficients of g and h. Then we have:

a n = n k=0 b k c n-k .
We also know

+∞ n=0 |b n | 2 = +∞ n=0 |c n | 2 = 1.
We want to show that

+∞ n=0 a n r n n + 2 2 √ n + 1 2 ≤ 1.
We can assume that the coefficients b n et c n are all non negative (At worst, the modulus of a n become bigger but we search a sufficient condition for the inequality so this is not a problem).

So the last inequality is equivalent to

+∞ n=0 n k=0 b k c n-k r n d n + 2 2 √ n + 1 ≤ 1 for all non negative sequence (d n ) with 2 -norm 1. This is equivalent to +∞ k,n=0 b k c n r n+k d n+k n + k + 2 2 √ n + k + 1 ≤ 1.
We will get this inequality as soon as +∞ k,n=0

r n+k n + k + 2 2 √ n + k + 1 d n+k 2 ≤ 1.
But for any j, we only have j + 1 ways to write this integer as the sum of two integers. So it suffices to prove the following inequality

+∞ j=0 r 2j (j + 2) 2 4 d 2 j ≤ 1.
And with the definition of (d n ), it will be true as soon as

r 2j (j + 2) 2 4 ≤ 1 for any j ≥ 0.
This latter inequality is clearly true for j = 0 for any r, so we just have to compute

r 0 = inf j≥1 2 j + 2 1/j .
We can check easily that x → ln(2/(x+2))

x is increasing on [1, +∞[ so we obtain

r 0 = 2 3 .
The following is obvious and is just a rewritting of the norms. Proof. If r ≤ 2 3 , it suffices to apply the three previous lemmas. Conversely assume that P r : B

1 (D) → B 2 (D) is bounded with norm 1. Let a ∈ R, we have 1 + arz 2 B 2 (D) = 1 + a 2 r 2 2 1 + az B 1 (D) = 1 + a 2 8 + •(a 2 ).
So we have

1 + a 2 r 2 2 ≤ 1 + a 2 8 •(a 2 ) 2 = 1 + a 2 4 + •(a 2 ).
And so r 2 ≤ 1 2 . Now we have another consequence of the preceding results, which will be used in the next section, and is similar to Prop.2.

Proposition 10. Let ε > 0. Then T ε : B 1 → B 2 is bounded.
Proof. We consider the following sequence of operators (we keep the notations of the preceding theorem)

S k : B 1 (D) -→ B 2 (D), f -→ P p -ε k (f )
where P r is the classical Poisson kernel. Indeed if we apply Prop.8 to this sequence of operators and to a Dirichlet series f of the form (1), we obtain

S(f )(s) = n=p α 1 1 •• p α k k ≥1 a n (p -s-ε 1 ) α1 • • (p -s-ε k ) α k = n=p α 1 1 •• p α k k ≥1 a n n -s-ε = T ε (f )(s).
We know from the preceding theorem that P r B 1 (D)→B 2 (D) ≤ 1 for r quite small and we obtain our result for T ε because p -ε k → 0 when k goes to infinity and so the infinite product of the norm is finite. (i) The identity from H p to B 2p is bounded with norm 1.

Notations. Let

But, (ii) the identity from H p to B p is not compact. Actually, it is not a strictly singular operator.

Proof. (i) Recall ( [START_REF] Duren | Bergman spaces[END_REF]) that the identity from H p (D) to B 2p (D) is bounded with norm 1 so it suffices to use Prop.9 to get the boundedness of our operator.

(ii) In [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF], it is shown that H p P = H 2 P and by the same way we obtain that B p P = B 2 P . But clearly H 2 P = B 2 P so H p and B p have isomorphic infinitedimensional closed subspaces and so the identity from H p to B p is not strictly singular.

Remarks.

(i) When p = 1, (i) has already been proved by Helson (see [START_REF] Helson | Dirichlet series[END_REF]).

(ii) We can check easily that for every n = m we have:

e pn -e pm B p ≥ e pn B p = 2 p + 2 1/p
and then we obtain another proof of the non compactness in Th.12(ii).

(iii) Let us mention that it is immediate (without invoking Th.12(ii)) that the identity from H p to B 2p is not compact: indeed if it were, by restriction to the variable z 1 = 2 -s , the identity from H p (D) to B 2p (D) would be compact but this is not the case.

Actually, we can prove that H 2 ⊂ B 4 by a simple computation on the coefficients of the Dirichlet series. Let f be a Dirichlet series of the form (1). We want to show that f B 4 ≤ f H 2 . We have

f 4 B 4 = f 2 2 B 2 . But f 2 (s) = +∞ n=1 b n n -s with b n = d|n a d × a n/d ∀n ≥ 1. So f 2 2 B 2 = +∞ n=1 | d|n a d × a n/d | 2 d(n) .
Now we apply the Cauchy-Schwarz inequality using the fact that the sum contains exactly d(n) terms

≤ +∞ n=1 d|n |a d | 2 |a n/d | 2 .
We have n ≥ 1 and n = d × n d , then we can exchange the sums

= +∞ d=1 |a d | 2 d|n |a n/d | 2 . But if n is a multiple of d, then n d is in N, and d|n |a n/d | 2 = f 2 H 2 .
At last, we get

f 4 B 4 ≤ f 4 H 2 .
D Generalized vertical limit functions.

Definition 9. Let χ ∈ D ∞ and f of the form (1). We denote by f χ the following Dirichlet series:

f χ (s) = +∞ n=1 a n χ(n)n -s .
In this part, we apply the same trick than in [START_REF] Gordon | The composition operators on the space of Dirichlet series with square summable coefficients[END_REF] and [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF] to obtain another expression of the norm in B p useful for the study of composition operators.

Let ϕ 1 (z) = 1+z 1-z the Cayley transform which maps D on C + . We will say that a function is in Proof. The Kronecker flow (T t ) is just a rotation on D ∞ , so

H p i (C + ) if f • ϕ 1 ∈ H p
D ∞ |g χ (it)| p dm(χ) = D ∞ |D(f )(T t χ)| p dm(χ) = D ∞ |D(f )(χ)| p dm(χ) = D(f ) p B p .
We conclude using the Fubini's theorem.

Proposition 11. Let χ ∈ D ∞ and f ∈ B p . Then g χ ∈ H p i (C + ) and g χ is an extension of f χ on C + .

Proof. Thanks to the previous lemma, we already know that for almost every χ ∈ D ∞ , g χ ∈ L p (λ i ). So it suffices to show (we use here a characterisation of the classical Hardy space) ∞ -∞ 1 -it 1 + it n g χ (it)dλ i (t) = 0 for n ≥ 1.

We use the same ideas than in [START_REF] Gordon | The composition operators on the space of Dirichlet series with square summable coefficients[END_REF] and [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF] but we have to adapt the proof because here we do not work with Fourier series on T ∞ but with functions in B p (D ∞ ). We fix n ≥ 1 and define G(χ) by Then we define the analytic polynomial

G(χ) := ∞ -∞ 1 -it 1 + it n g χ (it)dλ i (t). Clearly G ∈ L p (D ∞ ) because D ∞ |G(χ)| p dm(χ) = D ∞ +∞ -∞ 1 -it 1 + it n g χ (it)dλ i (t)
Q k (χ) := +∞ -∞ 1 -it 1 + it n P k (T t χ)dλ i (t)
and we claim that (Q k ) converges to G. Indeed

G -Q k p B p (D ∞ ) = D ∞ +∞ -∞ 1 -it 1 + it n
(g χ (it) -P k,χ (it)) dλ i (t) p dm(χ).

We get, through the Fubini's theorem,

G -Q k p B p (D ∞ ) ≤ +∞ -∞ (D(f ) -P k )(T t (•)) p B p (D ∞ ) dλ i (t)
but T t is just a rotation:

G -Q k p B p (D ∞ ) ≤ +∞ -∞ (D(f ) -P k ) p B p (D ∞ ) dλ i (t) = (D(f ) -P k ) p B p (D ∞ )
which goes to zero when k goes to infinity, and this proves our claim.

We claim now that G vanishes almost everywhere. Since G belongs to B p (D ∞ ), it suffices to prove that G is orthogonal to every monomial with positive index. Let q ∈ N. We have

D ∞ χ(q)G(χ)dm(χ) = +∞ -∞ 1 -it 1 + it n D ∞
χ(q)g χ (it)dm(χ) dλ i (t).

Actually we have D ∞ χ(q)g χ (it)dm(χ) = 0 because g χ (it) = Df (T t χ) ∈ B p (D ∞ ). This is clear for the polynomials and by density this proves the claim.

The proof that g χ is an extension of f χ is the same than in the case of H p (see [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF]). Now we shall denote f χ the extension instead of g χ . Like in the case of H p with p ≥ 1, this extension is almost surely simple. So the Menchoff's lemma gives that c n Φ n (χ) converges for almost every χ . Therefore, we get the result when p = 2.

When p = 2, it suffices to prove the result for p = 1. As in the case of the spaces A p , the result follows from Prop.10.

Let f ∈ B 2 , we know that for almost all χ ∈ D ∞ , f χ converges on C + and so g χ = f χ , we obtain for each probability measure w on R:

f 2 B 2 = R D ∞
|f χ (it)| 2 dA(χ)dw(t).

Theorem 13. Let f ∈ B 2 and w be a probability measure on R. Then Proof. For σ > 0, we have

D ∞ R |f χ (σ + it)| 2 dw(t)dm(χ) = f 2 B 2 = +∞ n=2 |a n | 2 n -2σ log 2 (n) d(n) .

1 )

 1 is uniformly convergent for (s) > a} = abscissa of uniform convergence of f. σ b (f ) = inf{a | The function f has an analytic, bounded extension for (s) > a} = abscissa of boundedness of f.

Theorem 3 .

 3 Let p ≥ 1 and z ∈ D. The point evaluation at z is bounded on A p σ and we have

2 ∀n, m ≥ 1 . 5 . 1 .

 2151 Definition Let m ≥ 1 be an integer. We define d m as the following multiplicative function d m (k) = d1...dm=k d1,..., d k ≥1 Remark. If we denote * the Dirichlet convolution, d m is multiplicative because d m = 1I * • • • * 1I (m times) where 1 I(n) = 1 for every n ≥ 1.

Proposition 7 .

 7 Let p ≥ 1.

Lemma 3 .a n ρ n+1 r n e inθ dρ dθ ≤ 1 aLemma 4 .

 314 The sequence 2 n + 2 n≥0 defines a multiplier fromB 1 (D) to H 1 (D)with norm exactly equal to 1:for every f (z) = n≥1 a z n ∈ B 1 (D), we have n≥1 2 n + 2 a n z n H 1 (D) ≤ f B 1 (D) . Proof. Let r < 1 and f ∈ B 1 (D) of the form f (z) = +∞ n=0a n z n . Then if we denote by M this multiplier operator, we have n ρ n r n e inθ ρ dρdθ.If r goes to 1, we obtain the result. Let r ≤ 2 3 . Then r n n+22 √ n+1 n≥0is a multiplier from H 1 (D) to H 2 (D) with norm 1.

Lemma 5. ( √ n + 1 ) 1 .

 11 n≥0 is a multiplier from H 2 (D) to B 2 (D) with norm exactly equal to Now we can state a contractive type result on classical Bergman spaces. Here P r denotes the blow-up operator: P r (f )(z) = f (rz).

Theorem 11 . 3 ,

 113 If r ≤ 2 P r : B 1 (D) → B 2 (D) is bounded with norm 1. Conversely, if P r : B 1 (D) → B 2 (D) is bounded with norm 1 then r ≤ 1 √ 2 .

Definition 10 .Lemma 6 .

 106 (D) (The classical Hardy space of the unit disk). In this case we have1 2π π -π |f • ϕ 1 (e iθ )| p dθ = R |f (it)| p dλ i (t)wheredλ i (t) = dt π(1 + t 2 ). Let t ∈ R. We define the Kronecker flow T t on D ∞ by T t (z 1 , z 2 , . . . ) := (p -it 1 z 1 , p -it 2 z 2 , . . . ). Let χ ∈ D ∞ , f ∈ B p and t ∈ R. We set g χ (it) := D(f )(T t χ). Then for w a finite Borel measure on R, one hasD ∞ R |g χ (it)| p dw(t)dm(χ) = w(R) f p B p .

  χ (it)| p dλ i (t)dm(χ) = D(f ) p B p where the last inequality follows from the preceding lemma. Actually, G ∈ B p (D ∞ ). It suffices to show that there exists a sequence of analytic polynomials which converges to G. Since f ∈ B p (D ∞ ), we have D(f ) ∈ B p (D ∞ ) and there exists a sequence (P k ) of analytic polynomials such that D(f ) -P k B p (D ∞ ) -→ k→+∞ 0.

Proposition 12 .

 12 Let χ ∈ D ∞ and f ∈ B p for p ≥ 1. Then for almost every χ (relatively to the measureA on D ∞ ), f χ converges on C + . Proof. Let f ∈ B 2 of the form (1). We consider L 2 (D ∞ , A) and the orthonormal sequence Φ n (χ) = d(n)χ(n). For σ > 0 and t ∈ R, let c n := a n n -σ-it d(n) • We point out that a n / d(n) n≥1 ∈ 2 and that n -σ log(n) n≥1 ∈ ∞ hence +∞ n=1|c n | 2 log 2 (n) < +∞.

f 2 B 2 = 4 R

 224 |f (+∞)| 2 + +∞ 0 D ∞ σ|f χ (σ + it)| 2 dA(χ)dσdw(t).

  p ≥ 1. We denote H p P (resp. B p P )) the following subspace of H p (resp. B p ):

	H p P = span(e k , k ∈ P)	H p	resp. B p P = span(e k , k ∈ P) B p	.
	Theorem 12. Let p ≥ 1.

z k .

K is well defined thanks to the condition on ζ and the fact that (K n ) converges pointwise to K.

Remark. We know that

Proposition 6. Let P be an analytic polynomial on

Proof. By the reproducing kernel property of the classical Bergman space used several times, we obtain that:

The Cauchy-Schwarz inequality gives the result.

With the previous proposition, we can extend by density the evaluation defined on the analytic polynomials for z ∈ D ∞ ∩ 2 . For every f ∈ B 2 (D ∞ ), we denote f (ζ) this extension and we have

Moreover the norm of the evaluation is exactly

Actually in [START_REF] Cole | Representing measures and Hardy spaces for the infinite polydisk algebra[END_REF], the authors proved (in a more general setting) that f is holomorphic on D ∞ ∩ 2 . We shall need the next lemma.

We multiply by σ and it suffices to remark that:

We shall give here some inequalities between the B p norm and some weighted p norms of the coefficient of the functions (as in Th.9). Theorem 14. Let p ≥ 1.

Proof. We do not give the details since it follows the same ideas as in the proof of Th.9.

When 1 ≤ p ≤ 2.

For every integer n = p α1 1 p α2 2 . . . ≥ 1 and f ∈ L p (D ∞ , dA), let us define

where z(n) = zα1 1 zα2 2 . . . We point out that, when P is a Dirichlet polynomial

. . We have in that case

The same interpolation argument gives the conclusion.

Annexe: around the norm of the point evaluation

We wish to present here a principle to compare (relatively to p) the norm of the point evaluation. We shall work in a rather general framework of subspaces of functions of some L p spaces. When one work on classical spaces of analytic functions (Hardy-Bergman spaces), this principle is useless, since one can essentially work with any power of a function (up to some standard tools). In the context of Dirichlet series, a big difficulty is the fact that we have no way to consider f α when α is not an integer (and f ∈ D). The following method can be helpful and gives very precise result in some particular cases.

In this section, we consider some subspaces X p ⊂ L p (Ω, ν) of functions on Ω, where ν is a probability measure on Ω and p ≥ 1. We assume that there exists some algebra P ⊂ ∩ p≥1 X p which is dense in each X p (think to the polynomials in many contexts).

We fix some ω ∈ Ω and we assume that the point evaluation f ∈ X p → f (ω) is bounded with norm N p .

Let us mention that the most often, thanks to the theory of reproducing kernels, the value of N 2 is known (and easy to get).

We give here several very simple observations which we used in this paper.

Proposition 13. With the previous notations,

(iii) Let m be an integer. We have for every p ≥ 1,

Proof. (i) Let f and g in P where f L q 1 = 1 and g L q 2 = 1. The product f g still belongs to P ⊂ X p and we have N p ≥ N p f g L p = N p f g X p ≥ |f (ω)|.|g(ω)|.

Taking now the upper bound relatively to f and to g, the first assertion follows.

(ii) is trivial.

(iii) Let us point out that, by an obvious induction, we have N p ≥ N q1 • • • N qr as soon as

• In particular, we can write


Amina Kimouche

Olivier Renault

Sayanti Samaddar

Clemens Winkelmann

Hervé Courtois

Olivier Fruchart

Johann Coraux
email: johann.coraux@grenoble.cnrs.fr

Modulating charge

would be desirable, intercalation may proceed differently due to the markedly different conditions.

In this article we address these two questions, by studying high quality graphene prepared by CVD on Ir(111) thin films. Iridium is one of those transition metals, like Cu, Pt, and Au, that weakly interact with graphene. 12 Unlike on the latter three metals however, most defects and notably grain boundaries can be avoided on Ir(111), 13,14 so that their role in intercalation 10 can be ruled out.

Correlating complementary microscopic analysis, we show that exposure to air leads to the formation of an ultrathin intercalated oxide modifying the weak graphene-Ir interaction and generating variations of the charge density in graphene. We find that a common kind of defect, local delaminations of graphene from its substrate, so-called wrinkles, forming during cool down to room temperature after CVD due to the mismatch of thermal expansion coefficient of graphene and its substrate, are pathways for a slow intercalation of oxygen species.

Under UHV, prior to exposure to air the graphene-covered regions of the sample exhibit a typical flat topography, except for wrinkles and a periodic corrugation associated with the graphene/Ir(111) moiré 12 (Supplementary Information (SI), Figure S1). The situation is markedly different after exposure to air, as shown by atomic force microscopy (AFM). Inspecting phase contrast AFM images (SI, Figure S2) allows one to identify graphene-free and graphene-covered regions. 15 Figure 1a reveals that (i) graphene-free regions (named A hereafter) protrude out of the surface, (ii) around graphene edges and along wrinkles, flat ribbons (named B hereafter) whose height is substantially larger (see below) than the Ir atomic step edges, are found. The moiré observed with scanning tunneling microscopy (STM) under UHV before air exposure is no more visible with STM. 16

Figure 1. Oxidation of iridium on graphene-free and graphene-covered regions. a) AFM image of graphene/Ir(111) after exposure to air, exhibiting Ir oxide on graphene-free regions (labeled "A") and intercalated between graphene and Ir, in the form of ribbons along the wrinkles in graphene (labeled "B"). b) Close-up of the region framed in a), as a function of time. c) Height profiles along the lines marked in b). "0" marks the position of the wrinkle. d) STM topograph of a graphene-covered region comprising a type-B region (right part). e) Cartoon summarizing the effect of atmospheric molecules (sketched with balls).

While the height (typically 3 nm in average, relative to graphene) of type A-regions does not evolve with time, the height of type B-regions, as well as their lateral expansion, does (Figure 1b,c).

After a few days the height and width of type B-regions do not increase much and reaches about 1.5 nm and a few 100 nm respectively. The surface of graphene appears rougher on type-B regions than on other graphene regions (Figure 1d). What is the nature of the height variations upon exposure to air? Volume expansion of surfaces due to their oxidation is a well-known effect in AFM. 17 The presence of type A-regions is interpreted in this light: in the absence of a protecting layer hindering surface air oxidation, an Ir oxide rapidly forms. 18,19 This oxide has a low conductivity, compared to the rest of the sample (SI, Figure S3). Type B-regions, which are covered with graphene, are tentatively identified at this point as Ir oxide ultrathin films progressively forming upon the reactive Final version of the manuscript at http://dx.doi.org/10.1016/j.carbon.2013.10.033 intercalation of oxygen.

In order to confirm this scenario, we performed spatially-resolved measurements of the work-function (WF). 20 PEEM spectroscopic images at the photoemission threshold exhibit a contrast inversion as a function of the secondary electron energy, indicative of a strong WF contrast across the surface (Figure 2a andS5). WF maps reveal two kinds of regions (Figure 2b,c): one with a 4-4.1 eV WF, the other with a mean 4.5 eV WF and variations of approximately 0.1 eV about this values across micrometer distances. The first value is close to the 4.2 eV one reported for IrO 2 , 18 and thus presumably corresponds to type A-regions in Figure 1.The second value is similar to that reported for graphene/Ir(111) (Ref. [21]) and thus signals graphene-covered regions, the spatial fluctuations in their WF seemingly corresponding to surface inhomogeneities, with length-scale compatible with the distance between intercalated (type B) ribbons.

Figure 2. Surface electronics and elemental composition. a) UV-excited PEEM spectroscopic image (4.4 eV secondary electrons) of a surface area comprising graphene-free (type A-regions) and graphene-covered regions, including type B-regions (oxide ribbon intercalated below graphene). b) Corresponding WF mapping of the region in a). c) WF distribution of b). d-g) Background-subtracted XPS micro-spectra of Ir, C, and O core levels over two areas of interest marked in a) (colored spectra are data obtained from the hexagonal frames in a) having the corresponding color; black curves are fits to the data).

The picture sketched above is further supported by chemical analysis of the regions with different WF, thanks to area-selected X-ray photoemission performed with the same PEEM set-up operated in the micro-spectroscopic mode. 20 We find that the intensity ratio between C 1s and Ir 4d peaks (Figure 2d,f) is 12 % in the high WF regions, and about twice as low on regions with prominently low WF. We note that due to the size of the iris aperture for area selection (sketched in Figure 2a), low WF regions also comprise a contribution (~ 30% here) of high WF regions. We deduce that low WF regions are graphene-free. An O 1s peak is detected in all regions (Figure 2e,g).

Though the statistics of our measurement does not allow to map out spatial inhomogeneities of this species, combining the information, gained from the WF and chemical analysis, we can conclude that low WF regions are graphene-free, type A-regions (IrO 2 ), and that high WF regions are graphene-covered and comprise oxygen species. The latter are inhomogeneously distributed over the surface, type B-regions (oxide ribbons intercalated along wrinkles), and possibly also, between these regions, a disordered layer of small oxygen-containing molecules (e.g. water, dioxygen) intercalated below graphene.

b) Raman spectra acquired on the two spots marked in c). c) Map of the position of graphene's 2D band Raman shift, in the same area as a). The grey region is graphene-free, doted lines are guides for the eyes showing the position of the wrinkles apparent in a). d) Local conductance (G) as a function of tip-sample bias (V bias ) averaged on the left region (red, type-B) region, and on the right (blue, graphene on Ir(111) without intercalated ribbon) of the STM image shown in inset.

At this point we summarize (Figure 1e) the effect of exposure to atmospheric environment at room temperature. Without holes in the graphene sheet, no oxidation of the Ir surface occurs.

Graphene-free regions are rapidly oxidized, on a time scale below 1 h. Graphene-covered regions are also oxidized, but on a much larger timescale, upon the slow intercalation of oxygen species at graphene edges entering wrinkles at their free end, which act as a tunnel for the transport of species across at least few micrometer distances. We note that the intercalation does not occur in case of high quality graphene fully covering the Ir(111) surface (as also noticed in Ref. [22]), i.e. when wrinkles do not have an open end.

In the remaining we focus on the type B-regions, more precisely on the influence of the intercalated oxide on the properties of graphene. Spatially-resolved Raman scattering measurements were performed at the exact location where AFM (Figure 3a) was also performed. No defectallowed D mode is observed (Figure 3b and S4), which confirms the uniform high quality of the graphene-covered regions. In the vicinity of the wrinkles, graphene's G and 2D modes are found at Final version of the manuscript at http://dx.doi.org/10.1016/j.carbon.2013.10.033 1588±3 and 2679±3 cm -1 respectively; they are located at 1594±3 and 2691±3 cm -1 far from the wrinkles (Figure 3b). Though the G and 2D mode softening, about 6 and 10 cm -1 , correlate with the position of the wrinkles (Figure 3c and S4), they are actually not a direct manifestation of graphene's deformation or decoupling from its support in the wrinkles. The G and 2D modes indeed have single-lorentzian profiles, in other words, they do not comprise two spectral components, which would each arise from "flat" and wrinkled graphene. A significant contribution from the wrinkles is actually not expected, given that the number of the atoms which they encompass is typically 100 times smaller than that illuminated by the laser spot (300 nm diameter). The observed blue-shifts are thus ascribed to the presence of the intercalated oxide ribbon. Strain and changes in electronic density can both be invoked in principle to explain G and 2D mode shifts. Rough estimates based on reports addressing specifically each effect yield order of magnitudes, 0.1 % and 10 12 cm -2 for strains 23 and electronic density change 24 respectively. The former could result from the stretching of graphene upon intercalation; 8 the latter could be associated with variations of the charge density in graphene set by the variation of the distance between graphene and the metal substrate due to the presence of the intercalated oxide. 16 Scanning tunneling spectroscopy was employed to test which effect (strain or charge transfer) actually prevails. The local density of state (LDOS, which is proportional to the local conductance measured with the STM tip) is markedly different on graphene regions with and without an intercalated oxide ribbon (Figure 3d). Concluding from the shape of the LDOS whether the graphene/support interaction is stronger or weaker is in our opinion daring, given the different shapes reported by different groups on identical graphene systems. 25,26 We shall only mention that the graphene/Ir interaction, which is known to be weak 12 can reasonably be expected to be even weaker in the presence of an interaction oxide, as is the case in graphene/Ru(0001). 9 Assuming that the LDOS minimum tracks the neutrality point in graphene, 27,28 we first observe that the Dirac point for graphene on Ir(111) is located closer to zero than in in situ UHV experiment. 12 The difference is ascribed to a transfer of electrons from below and above graphene. The latter may include the effect of adsorbates on graphene. The former includes the effect of a disordered layer of molecules possibly intercalated between graphene and Ir, the presence of which would explain both the occurrence of a strong Raman signal absent in as-grown graphene/Ir(111), 29 and the absence of the moiré pattern. Changes in the position of the Dirac point, of a few 100 meV, are observed between graphene with and without an oxide below it. This translates into variations of charge densities of the order of 10 12 cm -2 , which account for the observed Raman shift changes. We deduce that the latter originate prominently from charge density effects induced by intercalation.

In conclusion, we discovered that wrinkles play a central role in the diffusion of oxygen species in between graphene and its Ir(111) substrate under atmospheric conditions, by letting species penetrating below graphene from their open end. These diffusing species react with Ir and slowly form oxide ribbons whose thickness reaches 1.5 nm, expanding across a self-limited width of a few 100 nm. The effect was also observed recently in our laboratory on another related system, graphene on Cu foils, and is thus of general importance in CVD-produced graphene. We found strong evidence that the intercalation of the oxide ribbons yields modulation of few 10 12 cm -2 of the charge density in graphene and strongly modifies the graphene-support interaction. Our findings highlight a cheap and easy way of intercalating thin oxides between graphene and its substrate, and on the contrary, the crucial need for eliminating holes in graphene on metals with which it weakly interacts, in the view of any application taking benefit of graphene as a barrier coating to oxidation.

More advanced structures, with in-plane variations of the nature of the intercalants across 100 nm distances, could be designed by further processing. This opens the way to the bottom-up fabrication of high quality charge-modulated graphene, for instance in the view of smart devices exploiting the principles of optics for guiding charge carriers. 30,31

Experimental section

Final version of the manuscript at http://dx.doi.org/10.1016/j.carbon.2013.10.033

Graphene preparation: Graphene was prepared by CVD at 1300 K with ethylene, in a UHV system with 10 -10 mbar base pressure, on Ir(111) thin films previously deposited on C-plane sapphire wafers. 34 The growth was stopped after few minutes of exposure to a 10 -8 mbar partial pressure of ethylene in order to achieve a partial coverage of graphene. Microscopic characterizations: AFM was performed in tapping mode in atmospheric pressure.

Raman scattering measurement were done in atmospheric pressure with a WITec spectrometer coupled to an excitation laser with 532 nm wavelength and 2 mW/cm 2 power, and a 100× objective lens allowing a spatial resolution of about 300 nm. PEEM measurements were done under UHV after mild degassing (500 K) of the sample for 1 h, using a NanoESCA spectromicroscope (Omicron NanoTechnology). For the WF analysis, a UV Hg light source (4.9 eV) was used; the lateral resolution was better than 150 nm and the typical sensitivity of the WF determination was 20 meV. For the photoemission micro-spectra, a focused X-ray source (1486 eV) was illuminating the typical 35 µm field of view, and area selection over 5 µm-wide regions was performed thanks to a field aperture located in the first intermediate image plane of the PEEM optics; the overall energy resolution was 0.8 eV. STM/STS measurements: STM measurements were performed at room temperature. The spectra displayed are the average over 132 spectra, each obtained over 132 different points separated by 3.1 nm in each of the regions. STM images were acquired with 100 pA and 800 mV tunneling current and bias respectively.

Figure 3 .

 3 Figure 3. Change of the properties of graphene upon oxide intercalation. a) AFM image of graphene/Ir.b) Raman spectra acquired on the two spots marked in c). c) Map of the position of graphene's 2D band Raman shift, in the same area as a). The grey region is graphene-free, doted lines are guides for the eyes showing the position of the wrinkles apparent in a). d) Local conductance (G) as a function of tip-sample bias (V bias ) averaged on the left region (red, type-B) region, and on the right (blue, graphene on Ir(111) without intercalated ribbon) of the STM image shown in inset.

Final version of the manuscript at http://dx.doi.org/10.1016/j.carbon.2013.10.033

Acknowledgements

We thank Valérie Guisset and Philippe David for valuable support during the setting-up of the UHV experiments, and Nedjma Bendiab for fruitful discussions. Research supported by EU contract NMP3-SL-2010-246073 "GRENADA", and the French ANR contract ANR-2010-BLAN-1019-NMGEM.