Iteration of composition operators on small Bergman spaces of Dirichlet series

The Hilbert spaces $\mathscr{H}_{w}$ consisiting of Dirichlet series $F(s)=\sum_{ n = 1}^\infty a_n n^{ -s }$ that satisfty $\sum_{ n=1 }^\infty | a_n |^2/ w_n<\infty$, with $\{w_n\}_n$ of average order $\log_j n$ (the $j$-fold logarithm of $n$), can be embedded into certain small Bergman spaces. Using this embedding, we study the Gordon--Hedenmalm theorem on such $\mathscr{H}_w$ from an iterative point of view. By that theorem, the composition operators are generated by functions of the form $\Phi(s) = c_0s + \phi(s)$, where $c_0$ is a nonnegative integer and $\phi$ is a Dirichlet series with certain convergence and mapping properties. The iterative phenomenon takes place when $c_0=0$. It is verified for every integer $j\geqslant 1$, real $\alpha>0$ and $\{w_n\}_{n}$ having average order $(\log_j^+ n)^\alpha$ , that the composition operators map $\mathscr{H}_w$ into a scale of $\mathscr{H}_{w'}$ with $w_n'$ having average order $( \log_{j+1}^+n)^\alpha$. The case $j=1$ can be deduced from the proof of the main theorem of a recent paper of Bailleul and Brevig, and we adopt the same method to study the general iterative step.


INTRODUCTION
Let H 2 be the Hilbert space of Dirichlet series with square summable coefficients. A theorem of Gordon and Hedenmalm [5] classifies the set of analytic functions Φ : C 1/2 → C 1/2 which generate composition operators that map H 2 into itself. Let C θ denote the half-plane C θ := {s = σ + i t : σ > θ}. The Gordon-Hedenmalm theorem reads as follows, in a slightly strengthened version found in [10]. where c 0 is a nonnegative integer, and φ is a Dirichlet series that converges uniformly in C ǫ (ǫ > 0) with the following mapping properties: The set of such Φ is called the Gordon-Hedenmalm class and denoted by G . With the same convergence and mapping properties, the Gordon-Hedenmalm theorem was extended to the weighted Hilbert spaces D α which consists of Dirichlet series F (s) = ∞ n=1 a n n −s that satisfy ∞ n=1 |a n | 2 /d (n) α < ∞ in [2]. Here d (n) is the divisor function which counts the number of divisors of n and α > 0. In particular, for c 0 = 0, the composition operators map D α into D β with β = 2 α − 1. It should be noticed that D β are spaces that are smaller than D α when 0 < α < 1 and bigger when α > 1.
2010 Mathematics Subject Classification. 47B33, 30B50, 11N37. The author is supported by the Research Council of Norway grant 227768. 1 We observe from the proof of this extension (see [2,Theorem 1]) that D α are actually mapped into weighted Hilbert spaces that consist of Dirichlet series F (s) = ∞ n=1 a n n −s satisfying where Ω(n) is the number of prime factors of n (counting multiplicities). We say that an arithmetic function f has average order g if 1 X n X f (n) ≍ g (X ). Since d (n) α has average order (log n) β and Ω(n) β has average order (log log n) β (see Proposition 1), D α are in fact mapped into smaller weighted Hilbert spaces. In this paper, we show that the phenomenon of gaining one more fold of the logarithm persists for more general weights that have average order (log j n) α with j ∈ N and real α > 0. Let log j x denote the j -fold logarithm of x so that log 1 x = log x and log j x = log j −1 (log x). For convenience, we define log 0 x := x and log + |x| := max |x|, 0 ; log + j |x| := log + log j −1 + |x|, j 2.
Define (2) H w := F (s) = ∞ n=1 a n n −s : For every real number α > 0 and integer j 0, let H log,0 := F (s) = ∞ n=1 a n n −s : Our main result reads as follows. There are a few things we should make clear. First, it is proved in Section 4 that the average order of (log + j Ω(n)) α is (log + j +2 n) α , so that iterates of C Φ acting on H w fit into the scope of this theorem. Second, it is natural to replace D α with H w and w n = d α+1 (n) . Here, when α is a positive integer, d α (n) is the number of representations of n as a product of α integers, so d 2 (n) = d (n). For general α, d α (n) is the coefficient of the nth term of the Dirichlet series of ζ(s) α , i.e.
It can be checked that the proof of Theorem 1 of [2] carries through, so that C Φ maps H w with w n = d α+1 (n) into H Ω . Notice that d α+1 (n) has average order (log n) α [11] and d (n) α has average order (log n) 2 α −1 . 2

PRELIMINARIES
In [7], H 2 was identified with a space of analytic functions on D ∞ ∩ ℓ 2 (N), where D ∞ is the infinite polydisk D ∞ := z = (z 1 , z 2 , . . . ), |z j | < 1 . This is obtained by using the Bohr lift of Dirichlet series that are analytic in C 1/2 , which is defined in the following way. Let We write n as a product of its prime factors where the p j are the primes in ascending order. We replace p −s j with z j , set κ(n) = (k 1 , . . . , k r ), and define the formal power series as the Bohr lift of F .

Lemma 2.
Suppose that Φ ∈ G of the form (1) with c 0 = 0. For every Dirichlet polynomial F , every χ ∈ D ∞ and every s ∈ C 0 , we have Proof. It was proved in [2, Lemma 9] that whenever Φ ∈ G . This is reduced to (4) when c 0 = 0.
We shall now introduce a scale of Bergman spaces over D, as well as the corresponding Bergman spaces over C 1/2 which are induced by a certain conformal mapping τ : Let e j := exp(exp(· · · exp(e))) j e's (e 0 = 1). For α > 0 and j 1, we define Let D α,j (D) be the set of functions f that satisfy For f (z) = ∞ n=0 c n z n , we have and The proof of the main theorem will be given in Section 3. We verify that the average order of (log + j Ω(n)) α is (log + j +2 n) α in Section 4.

PROOF OF THEOREM 2
As in [2, Subsection 3.1], we inherit the proof of the arithmetical condition of c 0 from [5, Theorem A]. For the mapping and convergence properties of φ, we follow Subsection 3.2 in [2] as well.
Lemma 3. Assume that w n 1. There exists a function F ∈ H w such that (1) For almost all χ ∈ T ∞ , F χ converges in C 0 and cannot be analytically continued to any larger domain; (2) For at least one χ ∈ T ∞ , F χ converges in C 1/2 and cannot be analytically continued to any larger domain. 4 Proof. It was shown in [5] that the function (1) and (2). Clearly, F is in H w because w n 1.
The rest of the proof consists of two steps. We shall first embed H w into certain Bergman spaces, and then apply Littlewood's subordination principle to functions in these Bergman spaces.
Lemma 4 (Embedding of H w ). Let the weight {w n } of H w have average order (log j n) α . Then H w is continuously embedded into D α,j ,i (C 1/2 ).
For every τ ∈ Z, we define Q τ = (1/2, 1] × [τ, τ + 1). It suffices to prove the following local The case when j = 1 was established in [9]. We shall use the same method to establish the general case. It will suffice to prove the inequality We need the following lemma.

Lemma 5.
For α > 0 and j 2, letting n → ∞, we have Proof. We first prove the case j = 2 which is given by the integral We split the integral at the point t = 1/ log n, which is dictated by the exponential decay of the integrand. This gives I = I 1 + I 2 , where I 1 = For the general integral with j > 2 we can follow the same procedure. The main contribution comes from the term I 1 , and I 2 gives a negligible contribution, that is Proof of Lemma 4. Let F ∈ H w . Using duality, we have whereĝ is the Fourier transform of g . By the smoothness ofĝ and the assumption on w n , the supremum on the right hand side is finite. Integrating both sides against d µ * j (σ) and applying Lemma 5, we get the inequality (7).

THE AVERAGE ORDER
In this section, we will verify that the average order of (log + j Ω(n)) α is (log + j +2 n) α . It will suffice to give the details when j = 0.

Proposition 1.
For real α 1, we have This estimation is consistent with the case when α = 1 or 2 which can be found in [12]. Let We shall use some results of N k (X ) to estimate S α Ω (X ), for which we need to rewrite S α Ω (X ) as S α Applying Stirling's formula gives the sum X 2π log X log 2 X We now write k = log 2 X + ℓ with ℓ ∈ −ǫlog 2 X , ǫlog 2 X = − 2B log 2 X log 3 X , 2B log 2 X log 3 X so that Upon forming the product of these two series, our sum becomes for some coefficients c j . We also expand the first factor of our sum as follows Thus, our summand takes the form Upon performing the sum we only retain the terms with an even power of ℓ. Hence, we are led to compute On approximating the sum with an integral via the Euler-Maclaurin summation formula we gain an error term of order O 1 (log 2 X ) B . Then the leading term is given by For the higher powers of ℓ we use the formulae These give Putting everything together gives It should be clear from the above that with more precision one can obtain an asymptotic expansion to any required degree of accuracy. For (i) and (iii), we shall use the Erdős-Kac theorem for Ω(n) [11], which states that This gives = log 2 X α n X 1 Ω(n) (1−ǫ) log 2 X 1.
For (iv), by Nicolas's result in [8], there exists the same constant C , such that log X / log 2 We may bound this last sum from above by the integral log X / log 2 For the average order of log + j Ω(n) α , when carrying through the proof of Proposition 1 for the range (ii), it can be seen from (12) that the main contribution log j +2 X α gets more and more centralised when j becomes bigger. Therefore, we eventually get