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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 Fréchet 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.
Littlewood’s and Bohnenblust-Hille’s inequalities (recall Chapter 6) bound certain sequence norms of the coefficients of a polynomial by a constant (not depending on the number of variables) times the supremum of the polynomial on the polydisc. A similar problem is handled here, replacing the polydisc by the unit ball of C^n with some p-norm. Optimal exponents (that depend on the degree of the polynomial and on p) are given. The proof relies on the interplay between homogeneous polynomials and multilinear mappings and an analogous inequality for multilinear mappings. This one is proved by giving a generalized mixed inequality that bounds a mixed norm of the coefficients of a matrix by the supremum on the p-balls of the associated multilinear mapping.
The solution of Bohr’s problem (see Chapter 4) implies that for every Dirichlet series in \mathcal{H}_\infty, the sum ∑ |a_n| n^(-s) is finite for every Re s > 1/2, and we ask if we can in fact get to Re s=1/2. This is addressed by considering, for Dirichlet polynomials, the quotient between ∑ | a_n | and the norm (in \mathcal{H}_\infty) of the polynomial. We define S(x) as the supremum over all Dirichlet polynomials of length x ≥ 2 of these quotients. It is shown that S(x)=exp(- (1/\sqrt{2} + o(1)) (log n loglog n)^(1/2)) as x goes to ∞. This is reformulated in terms of the Sidon constant of the monomials as characters of the infinite-dimensional polydisc. The proof uses the hypercontractive Bohnenblust-Hille inequality and a fine decomposition of the natural numbers as those having ‘big’ and ‘small’ prime factors. Also, a version for homegeneous Dirichlet series is given.
We continue the study initiated in Chapter 7 of polynomials with small norms. This time the norm of the polynomial is not taken as the supremum on the n-dimensional polydisc, we take it on B_X, the unit ball of some Banach space. The goal is to show that, given a polynomial, signs can be found in such a way that the norm of the new polynomial, whose coefficients are the original ones multiplied by the signs, has small norm. We do this with three different approaches. The first two approaches use Rademacher random variables as the main probabilistic tools. The first one is based on finding out how many balls of a fixed radius are needed to cover B_X while the second one uses entropy integrals and a good estimate for the entropy numbers of the inclusions between l_p spaces. The third approach is different, and relies on Gaussian random variables, Slepian’s lemma and the fact that Rademacher averages are dominated by Gaussian averages. This approach also allows to get estimates for vector-valued polynomials.
This is a short introduction to the basics of the theory of normed tensor products. The m-fold tensor product of linear spaces is defined through the universal property. If the involved spaces are normed, then the projective and injective norms on the tensor product are. Basic properties are given: the metric mapping property and their relationship with continuous linear mappings. The symmetric m-fold tensor product and the symmetric projective and injective norms are defined analogously. These are related to the m-homogeneous polynomials.
Each Hardy space of Dirichlet series \mathcal{H}_p has an associated abscissa, and the analogue to Bohr’s problem arises in a natural way: to determine the maximal distance S_p between this abscissa and the abscissa of absolute convergence. If a Dirichlet series with coefficients (a_n) belongs to \mathcal{H}_p, then the series with coefficients (a_n/n^{ε}) belongs to \mathcal{H}_q for all q>p and ε >0. It is shown that S_p=1/2, and that, if we only consider m-homogeneous Dirichlet series, S_p^m=1/2. For every 1 ≤ p < ∞ the set of monomial convergence of the Hardy space H_p of functions on the infinite dimensional polytorus (hence also of the Hardy space H_2 on the infinite-dimensional polytorus) is l_2 ∩ Bc0. The space of all multipliers on the Hardy space of Dirichlet series \mathcal{H}_p coincides with \mathcal{H}_\infty.
We establish the basic notions around Dirichlet series that are going to be used all along the text. A Dirichlet series converges on half-planes, and that there it defines a holomorphic function. For a given Dirichlet series we consider four abscissas definining the maximal half-planes on which it: converges, defines a bounded holomorphic function, converges uniformly or converges absolutely. We formulate the problem of determining the maximal possible distance between these abscissas. The difference between the abscissa of convergence and absolute convergence is at most one, and this is attained. Also, the abscissa of uniform convergence and of boundedness always coincide (this is Bohr theorem). Then Bohr’s problem is established: to determine S, the maximal possible width of the strip of absolute but not uniform convergence of Dirichlet series, and we show that it is at most 1/2. Finally we introduce the Banach space \mathcal{H}_\infty of Dirichlet series that converge and define a bounded holomorphic function on the right half-plane and reformulate Bohr’s problem in terms of this space. This becomes later an important tool.
The text is closed by coming back to Bohr’s absolute convergence problem, this time for vector-valued Dirichlet series. For a Banach space X abscissas and strips S(X) and S_p(X), analogous to those defined in Chapters 1 and 12 are considered. It is shown that all these strips equal 1-1/cot(X), where cot(X) is the optimal cotype of X.
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^ε)) are in \mathcal{H}_p with uniformly bounded norms. Also, a version of the brothers Riesz theorem for vector-valued functions is given.
For each 1 ≤ p ≤ ∞, the Hardy space \mathcal{H}_p of Dirichlet series is defined as the image through the Bohr transform of the Hardy space of functions on the infinite-dimensional polytorus. The Dirichlet polynomials are dense in \mathcal{H}_p for every 1 ≤ p < ∞. For p=2 this coincides with the space of Dirichlet series whose coefficients are square-summable. A Dirichlet series with coefficients a_n belongs to\mathcal{H}_p if and only if the series with coefficients a_n/n^ε is in \mathcal{H}_p for every ε >0 and the norms are uniformly bounded. In this case, the series is the limit as ε tends to 0. As a technical tool to see this, vector-valued Dirichlet series (that is, series with coefficients in some Banach space) are introduced, and some basic definitions and properties (such as the convergence abscissas, Bohr-Cahen formulas) are given.
Given a family of formal power series, its set of monomial convergence is defined as those z’s for which the series converges. The main focus is given to the sets of monomial convergence of the m-homogeneous polynomials on c0 and of the bounded holomorphic functions on B_{c0}. The first one is completely described in terms of the Marcinkiewicz space l_{(2m)/(m-1), ∞}. For the second one there is no complete description. If z is such that limsup (log n)^(1/2) ∑_j^n (z*_j)^{2} < 1 (where z* is the decreasing rearrangement of z), then z is in the set of monomial convergence of the bounded holomorphic functions. Also, if z belongs to the set of monomial convergence, then the limit superior is ≤ 1. This is related to Bohr’s problem (see Chapter 1). First of all, if M denotes the supremum over all q so that l_q is contained in the set of monomial convergence of the bounded holomorphic functions on Bc0, then S=1/M. But this can be more precise: S is the infimum over all σ >0 so that the sequence (p_n^(-σ))_n (being p_n the n-th prime number) belongs to the set of monomial convergence of the bounded holomorphic functions on Bc0.
We work with integrable functions on the polytorus, both in finite and infinitely many variables. For such a function and a multi-index the corresponding Fourier coefficient is defined. For each 1 ≤ p ≤ ∞ the Hardy space H_p consists of those functions in L_p having non-zero Fourier coefficients only for multi-indices in the positive cone. The Hardy space H_\infty on the infinite dimensional polytorus and the space of bounded holomorphic functions on Bc0 are isometrically isomorphic. To prove this the Poisson kernel in several variables is defined, and the Poisson operator (defined through convolution with this kernel) is considered. With these it is shown that the trigonometric polynomials are dense in L_p for 1 ≤ p < ∞ and weak*-dense in L_\infty, and that so also are the analytic trigonometric polynomials in H_p and H_∞. The isometry between the two spaces is first established for the finite dimensional polytorus/polydisc and then, using a version of Hilbert’s criterion (see Chapter 2), raised to the infinite-dimensional case. The density of the polynomials can be proved using the Féjer kernel instead of the Poisson one.
The Bohr radius for p-norms was introduced and studied in Chapter 19. There it was shown that unconditional basis constants of the monomials in spaces of m-homogeneous polynomials and Bohr radii are, in a certain sense, reciprocal to each other. In Chapter 21 the Gordon-Lewis cycle of ideas was developed to study these unconditional basis constants. Relating unconditional basis constants, Gordon-Lewis constants and projection constants of spaces of m-homogeneous polynomials gives a new proof of the lower bound for the Bohr radius for p-norms.