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[EN] We study Hausdorff-Young-type inequalities for vector-valued Dirichlet series which allow us to compare the norm of a Dirichlet series in the Hardy space H-p(X) with the q-norm of its coefficients. In order to obtain inequalities completely analogous to the scalar case, a Banach space must satisfy the restrictive notion of Fourier type/cotype. We show that variants of these inequalities hold for the much broader range of spaces enjoying type/cotype. We also consider Hausdorff-Young-type inequalities for functions defined on the infinite torus T-infinity or the boolean cube {- 1, 1}(infinity). As a fundamental tool we show that type and cotype are equivalent to a hypercontractive homogeneous polynomial type and cotype, a result of independent interest., The third author was supported by MICINN and FEDER Project MTM2017-83262-C2-1-P and MECD grant PRX17/00040.

[EN] Inspired by a recent article on Frechet spaces of ordinary Dirichlet series Sigma a(n)n(-s) due to J. Bonet, we study topological and geometrical properties of certain scales of Frechet spaces of general Dirichlet spaces Sigma a(n)e(-lambda ns) focus on the Frechet space of lambda-Dirichlet series Sigma a(n)e(-lambda ns) which have limit functions bounded on all half planes strictly smaller than the right half plane [Re > 0]. We develop an abstract setting of pre-Frechet spaces of lambda-Dirichlet series generated by certain admissible normed spaces of lambda-Dirichlet series and the abscissas of convergence they generate, which allows also to define Frechet spaces of lambda-Dirichlet series for which a(n)e(-lambda n/k) for each k equals the Fourier coefficients of a function on an appropriate lambda-Dirichlet group., Andreas Defant: Partially supported by MINECO and FEDER project MTM2017-83262-C2-1-P. Tomás Fernández Vidal: Supported by PICT 2015-2299. Pablo Sevilla-Peris:Supported by MINECO and FEDER project MTM2017-83262-C2-1-P

We study composition operators on spaces of double Dirichlet series, focusing our interest on the characterization of the composition operators of the space of bounded double Dirichlet series $${\mathcal {H}}^\infty ({\mathbb {C}}_+^2)$$ . We also show how the composition operators of this space of Dirichlet series are related to the composition operators of the corresponding spaces of holomorphic functions. Finally, we give a characterization of the superposition operators in $${\mathcal {H}}^\infty ({\mathbb {C}}_+)$$ and in the spaces $${\mathcal {H}}^p$$ .

[EN] We study mean ergodic composition operators on infinite dimensional spaces of holomorphic functions of different types when defined on the unit ball of a Banach or a Hilbert space: that of all holomorphic functions, that of holomorphic functions of bounded type and that of bounded holomorphic functions. Several examples in the different settings are given., We are very grateful to Jose Bonet for valuable suggestions about this work and to Pablo Galindo for pointing out an example of a Banach space which satisfies Corollary5.7. We also thank the referee for her/his careful reading and useful suggestions. The research of the first author was partially supported by the project MTM2016-76647-P. The research of the second author was partially supported by the project GV Prometeo 2017/102. The research of the third author was supported by the project MTM2017-83262-C2-1-P.

[EN] We study the Hardy space of translated Dirichlet series H+. It consists on those Dirichlet series Sigma a(n)n(-s) such that for some (equivalently, every) 1 0. We prove that this set, endowed with the topology induced by the seminorms {parallel to center dot parallel to(2,k)}(k is an element of N) (where parallel to Sigma a(n)n(-s) parallel to(2,k) is defined as parallel to Sigma a(n)n(-(s+ 1/k))parallel to(H2)), is a Frechet space which is Schwartz and non nuclear. Moreover, the Dirichlet monomials {n(-s)}(n is an element of N) are an unconditional Schauder basis of H+. We also explore the connection of this new space with spaces of holomorphic functions on infinite-dimensional spaces. In the spirit of Gordon and Hedenmalm's work, we completely characterize the composition operator on the Hardy space of translated Dirichlet series. Moreover, we study the superposition operators on H+ and show that every polynomial defines an operator of this kind. We present certain sufficient conditions on the coefficients of an entire function to define a superposition operator. Relying on number theory techniques we exhibit some examples which do not provide superposition operators. We finally look at the action of the differentiation and integration operators on these spaces., We would like to warmly thank Jose Bonet, Andreas Defant and Manuel Maestre for enlightening remarks and comments and fruitful discussions that improved the paper. We would also like to thank the referees for their careful reading and helpful comments. The research of T. Fernandez Vidal was supported by PICT 2015-2299. The research of D. Galicer was partially supported by CONICET-PIP 11220130100329CO and 2018-04250. The research of M. Mereb was partially supported by CONICET-PIP 11220130100073CO and PICT 2018-03511. The research of P. Sevilla-Peris was supported by MICINN and FEDER Project MTM2017-83262-C2-1-P and MECD Grant PRX17/00040.

A classical result of Fatou gives that every bounded holomorphic function on the disc has radial limits for almost every point in the torus, and the limit function belongs to the Hardy space H_\infty of the torus. This property is no longer true when we consider vector-valued functions. The Banach spaces X for which this property is satisfied are said to have the analytic Radon-Nikodym property (ARNP). Some important equivalent reformulations of ARNP are studied in this chapter. Among others, X has ARNP if and only if each X-valued H_p- function f on the disc has radial limits almost everywhere on the torus (and not only H_\infty-functions). Even more, in this case each such f has non-tangential limits within any Stolz region. The basic tools are subharmonic functions and certain maximal inequalities. Finally, it is shown that if X has the ARNP, then every L_p of functions taking values in X with a finite measure also has ARNP.

Given a Banach space X, we consider Hardy spaces of X-valued functions on the infinite polytorus, Hardy spaces of X-valued Dirichlet series (defined as the image of the previous ones by the Bohr transform), and Hardy spaces of X-valued holomorphic functions on l_2 ∩ B_{c0}. The chapter is dedicated to study the interplay between these spaces. It is shown that the space of functions on the polytorus always forms a subspace of the one of holomorphic functions, and these two are isometrically isomorphic if and only if X has ARNP. Then the question arises of what do we find in the side of Dirichlet series when we look at the image of the Hardy space of holomorphic functions. This is also answered, showing that this consists of Dirichlet series for which all horizontal translations (those whose coefficients are (a_n/n^e)) are in \mathcal{H}_p with uniformly bounded norms. Also, a version of the brothers Riesz theorem for vector-valued functions is given.

This is a short introduction to the theory of holomorphic functions in finitely and infinitely many variables. We begin with functions in finitely many variables, giving the definition of holomorphic function. Every such function has a monomial series expansion, where the coefficients are given by a Cauchy integral formula. Then we move to infinitely many variables, considering functions defined on B_{c0}, the open unit ball of the space of null sequences. Holomorphic functions are defined by means of Frechet differentiability. We have versions of Weierstrass and Montel theorems in this setting. Every holomorphic function on B_{c0} defines a family of coefficients through a Cauchy integral formula and a (formal) monomial series expansion. Every bounded analytic (represented by a convergent power series) function is holomorphic. Hilbert’s criterion, that gives conditions on a family of scalars so that it is attached to a bounded holomorphic function on B_{c0}. Homogeneous polynomials are those entire functions having non-zero coefficients only for multi-indices of a given order. We show how these are related to multilinear forms on c0 through the polarization formulas.

[EN] A classical result of Harald Bohr linked the study of convergent and bounded Dirichlet series on the right half plane with bounded holomorphic functions on the open unit ball of the space c(0) f complex null sequences. Our aim here is to show that many questions in Dirichlet series have very natural solutions when, following Bohr's idea, we translate these to the infinite dimensional setting. Some are new proofs and other new results obtained by using that point of view., The authors were supported by MINECO and FEDER Project MTM2017-83262-C2-1-P. The second and third authors were also supported by Project Prometeo/2017/102 of the Generalitat Valenciana.

We employ a classical result by Toeplitz (1913) and the seminal work by Bohnenblust and Hille on Dirichlet series (1931) to show that the set of continuous m-homogeneous non-analytic polynomials on c0 contains an isomorphic copy of l1. Moreover, we can have this copy of l1 in such a way that every non-zero element of it fails to be analytic at precisely the same point.

[EN] The Cluster Value Theorem is known for being a weak version of the classical Corona Theorem. Given a Banach space X, we study the Cluster Value Problem for the ball algebra A(u)(B-X), the Banach algebra of all uniformly continuous holomorphic functions on the unit ball B-X; and also for the Frechet algebra H-b(X) of holomorphic functions of bounded type on X (more generally, for H-b(U), the algebra of holomorphic functions of bounded type on a given balanced open subset U subset of X). We show that Cluster Value Theorems hold for all of these algebras whenever the dual of X has the bounded approximation property. These results are an important advance in this problem, since the validity of these theorems was known only for trivial cases (where the spectrum is formed only by evaluation functionals) and for the infinite dimensional Hilbert space. As a consequence, we obtain weak analytic Nullstellensatz theorems and several structural results for the spectrum of these algebras. (C) 2017 Elsevier Inc. All rights reserved., This work was partially supported by projects CONICET PIP 11220130100329CO, ANPCyT PICT 2015-2299, ANPCyT PICT-2015-3085, ANPCyT PICT-2015-2224, UBACyT 20020130300057BA, UBACyT 20020130300052BA, UBACyT 20020130100474BA and MINECO and FEDER Project MTM2014-57838-C2-2-P.

[EN] We present an abstract approach to the abscissas of convergence of vector-valued Dirichlet series. As a consequence we deduce that the abscissas for Hardy spaces of Dirichlet series are all equal. We also introduce and study weak versions of the abscissas for scalar-valued Dirichlet series., A. Defant: Partially supported by MINECO and FEDER MTM2017-83262-C2-1-P. A. Pérez: Supported by La Caixa Foundation, MINECO and FEDER MTM2014-57838-C2-1-P and Fundación Séneca - Región de Murcia (CARM 19368/PI/14). P. Sevilla-Peris: Supported by MINECO and FEDER MTM2017-83262-C2-1-P.

[EN] We estimate the -norm of finite Dirichlet polynomials with coefficients in a Banach space. Our estimates quantify several recent results on Bohr's strips of uniform but non absolute convergence of Dirichlet series in Banach spaces., A. Defant and P. Sevilla-Peris were supported by MICINN Project MTM2011-22417.

[EN] We introduce non-linear versions of the classical cotype of Banach spaces. We show that spaces with l.u.st. and cotype, and spaces having Fourier cotype enjoy our non-linear cotype. We apply these concepts to get results on convergence of vector-valued power series in infinite many variables and on l(1)-multipliers of vector-valued Dirichlet series. Finally we introduce cotype with respect to indexing sets, an idea that includes our previous definitions. (C) 2015 Elsevier Inc. All rights reserved., The first author was partially supported by CONICET-PIP 11220130100329CO, UBACyT 20020130100474BA and ANPCyT PICT 2011-1456. The third author was also supported by UPV-SP20120700. All three authors were supported by project MTM2014-57838-C2-2-P.

We unify Littlewood's classical 4/3-inequality (a forerunner of Grothendieck's inequality) together with its m -linear extension due to Bohnenblust and Hille (which originally settled Bohr's absolute convergence problem for Dirichlet series) with a scale of inequalties of Bennett and Carl in l p -spaces (which are of fundamental importance in the theory of eigenvalue distribution of power compact operators). As an application we give estimates for the monomial coefficients of homogeneous l p -valued polynomials on c 0 .

By the von Neumann inequality for homogeneous polynomials there exists a positive constant $C_{k,q}(n)$ such that for every $k$-homogeneous polynomial $p$ in $n$ variables and every $n$-tuple of commuting operators $(T_1, \dots, T_n)$ with $\sum_{i=1}^{n} \Vert T_{i} \Vert^{q} \leq 1$ we have \[ \|p(T_1, \dots, T_n)\|_{\mathcal L(\mathcal H)} \leq C_{k,q}(n) \; \sup\{ |p(z_1, \dots, z_n)| : \textstyle \sum_{i=1}^{n} \vert z_{i} \vert^{q} \leq 1 \}\,. \] For fixed $k$ and $q$, we study the asymptotic growth of the smallest constant $C_{k,q}(n)$ as $n$ (the number of variables/operators) tends to infinity. For $q = \infty$, we obtain the correct asymptotic behavior of this constant (answering a question posed by Dixon in the seventies). For $2 \leq q < \infty$ we improve some lower bounds given by Mantero and Tonge, and prove the asymptotic behavior up to a logarithmic factor. To achieve this we provide estimates of the norm of homogeneous unimodular Steiner polynomials, i.e. polynomials such that the multi-indices corresponding to the nonzero coefficients form partial Steiner systems., 14 pages, Accepted in Journal f\"ur die reine und angewandte Mathematik (Crelle's Journal)

[EN] The Bohr–Bohnenblust–Hille theorem states that the width of the strip in the complex plane on which an ordinary Dirichlet series P n ann −s converges uniformly but not absolutely is less than or equal to 1 2 , and this estimate is optimal. Equivalently, the supremum of the absolute convergence abscissas of all Dirichlet series in the Hardy space H∞ equals 1 2 . By a surprising fact of Bayart the same result holds true if H∞ is replaced by any Hardy space Hp, 1 ≤ p < ∞, of Dirichlet series. For Dirichlet series with coefficients in a Banach space X the maximal width of Bohr’s strips depend on the geometry of X; Defant, García, Maestre and Pérez-García proved that such maximal width equals 1 − 1/Cot X, where Cot X denotes the maximal cotype of X. Equivalently, the supremum over the absolute convergence abscissas of all Dirichlet series in the vector-valued Hardy space H∞(X) equals 1 − 1/Cot X. In this article we show that this result remains true if H∞(X) is replaced by the larger class Hp(X), 1 ≤ p < ∞, Carando was partially supported by CONICET PIP 0624, PICT 2011-1456 and UBACyT 1-746. Defant and Sevilla-Peris were supported by MICINN project MTM2011-22417. Sevilla-Peris was partially supported by UPV-SP20120700.

Let $\mathcal{H}_\infty$ be the set of all ordinary Dirichlet series $D=\sum_n a_n n^{-s}$ representing bounded holomorphic functions on the right half plane. A multiplicative sequence $(b_n)$ of complex numbers is said to be an $\ell_1$-multiplier for $\mathcal{H}_\infty$ whenever $\sum_n |a_n b_n| < \infty$ for every $D \in \mathcal{H}_\infty$. We study the problem of describing such sequences $(b_n)$ in terms of the asymptotic decay of the subsequence $(b_{p_j})$, where $p_j$ denotes the $j$th prime number. Given a multiplicative sequence $b=(b_n)$ we prove (among other results): $b$ is an $\ell_1$-multiplier for $\mathcal{H}_\infty$ provided $|b_{p_j}| < 1$ for all $j$ and $\overline{\lim}_n \frac{1}{\log n} \sum_{j=1}^n b_{p_j}^{*2} < 1$, and conversely, if $b$ is an $\ell_1$-multiplier for $\mathcal{H}_\infty$, then $|b_{p_j}| < 1$ for all $j$ and $\overline{\lim}_n \frac{1}{\log n} \sum_{j=1}^n b_{p_j}^{*2} \leq 1$ (here $b^*$ stands for the decreasing rearrangement of $b$). Following an ingenious idea of Harald Bohr it turns out that this problem is intimately related with the question of characterizing those sequences $z$ in the infinite dimensional polydisk $\mathbb{D}^\infty$ (the open unit ball of $\ell_\infty$) for which every bounded and holomorphic function $f$ on $\mathbb{D}^\infty$ has an absolutely convergent monomial series expansion $\sum_{\alpha} \frac{\partial_\alpha f(0)}{\alpha!} z^\alpha$. Moreover, we study analogous problems in Hardy spaces of Dirichlet series and Hardy spaces of functions on the infinite dimensional polytorus $\mathbb{T}^\infty$., Comment: arXiv admin note: substantial text overlap with arXiv:1207.2248

[EN] In 1931 H.F.Bohnenblust and E.Hille published a very important paper in which not only did they solve a long standing problem on convergence of Dirichlet series, but also gave a general version of a celebrated inequality of Littlewood. Although it is full of extremely valuable mathematical ideas, the paper has been overlooked for a long time and even today we feel that it does not get the credit it deserves. This may be caused by the not always accessible style that makes that the ideas are sometimes hidden. It is our intention to try to study the paper from a modern point of view and to bring to light the valuable aspects we believe it has., Both authors were supported by MICINN Project MTM2011-22417. The second author was also partially supported by project UPV-SP201207000.

[EN] Hartman proved in 1939 that the width of the largest possible strip in the complex plane on which a Dirichlet series is uniformly a.s.- sign convergent (i.e., converges uniformly for almost all sequences of signs epsilon (n) = +/- 1) but does not convergent absolutely, equals 1/2. We study this result from a more modern point of view within the framework of so-called Hardytype Dirichlet series with values in a Banach space., Supported by CONICET-PIP 11220130100329CO, PICT 2015-2299 and UBACyT 20020130100474BA. Supported by MICINN MTM2017-83262-C2-1-P. Supported by MICINN MTM2017-83262-C2-1-P and UPV-SP20120700.

[EN] Denote by Ω(n) the number of prime divisors of n ∈ N (counted with multiplicities). For x ∈ N define the Dirichlet-Bohr radius P L(x) to be the best r > 0 such that for every finite Dirichlet polynomial n≤x ann −s we have X n≤x |an|r Ω(n) ≤ sup t∈R X n≤x ann −it . We prove that the asymptotically correct order of L(x) is (log x) 1/4x −1/8 . Following Bohr’s vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows to translate various results on Bohr radii in a systematic way into results on Dirichlet-Bohr radii, and vice versa, The first author was supported by CONICET-PIP 0624, UBACyT 20020130100474BA and ANPCyT PICT 2011-1456. The third, fourth and fifth authors were supported by MINECO MTM2014-57838-C2-2-P, and the third and fourth also by Prometeo 11/2013/013. The fifth author was also partially supported by UPV-SP20120700.

[EN] Each Dirichlet series D=∑∞n=1an1nsD=∑n=1∞an1ns, with variable s∈Cs∈C and coefficients an∈Can∈C, has a so called Bohr strip, the largest strip in CC on which DD converges absolutely but not uniformly. The classical Bohr-Bohnenblust-Hille theorem states that the width of the largest possible Bohr strip equals 1/21/2. Recently, this deep work of Bohr, Bohnenblust and Hille from the beginning of the last century was revisited by various authors. New methods from different fields of modern analysis (e.g. probability theory, number theory, functional and Fourier analysis) allow to improve the Bohr-Bohnenblust-Hille cycle of ideas, and to extend it to new settings, in particular to Dirichlet series which coefficients in Banach spaces. We survey on various aspects of these new developments., The authors were supported in part by MICINN and FEDER Project MTM2008-03211. The third author was also supported by Prometeo 2008/101

[EN] Recent results on Dirichlet series Sigma(n) a(n) 1/n(s), s is an element of C, with coefficients a(n) in an infinite dimensional Banach space X show that the maximal width of uniform but not absolute convergence coincides for Dirichlet series and for m-homogeneous Dirichlet polynomials. But a classical non-trivial fact fue to Bohnenblust and Hille shows that if X is one dimensional, this maximal width heavily depends on the degree m of the Dirichlet polynomials. We carefully analyze this phenomenon, in particular in the setting of l(p)-spaces., Both authors were supported by the MEC Project MTM2008-03211 The second author was partially supported by grants PR2007 0384 (MEC) and UPV PAID 00-07

We give asymptotically correct estimations for the cotype 2 constant C2(P(mXn)) ofthe spaceP(mXn) of allm-homogenous polynomials onXn, the span of the firstn sequencesek=(\gdkj)j in a Banach sequence spaceX. Applications to Minkowski, Orlicz and Lorentz sequence spaces are given.