New strips of convergence for Dirichlet series. (English) Zbl 1207.30001

Defant, Andreas; García, Domingo; Maestre, Manuel

Publ. Mat., Barc. 54, No. 2, 369-388 (2010).

The authors study the interplay of the theory of classical Dirichlet series in one complex variable with recent developments on monomial expansions of holomorphic functions in infinitely many variables (from the abstract).
More detailed: For a Dirichlet series \(\sum_1^\infty \frac{a_n}{n^s}\) the abscissa of absolute (resp. uniform) convergence is \[ \begin{matrix} \sigma_a\\ \sigma_u\end{matrix} = \inf_{r\in\mathbb R} \left\{ \sum_1^\infty \frac{a_n}{n^s}\quad\begin{matrix}\text{absolutely}\\ \text{uniformly}\end{matrix}\text{ convergent on }\operatorname{Re}(s) \geq r\right\}. \] H. Bohr was interested in \(T\), the maximum of the differences \(\sigma_a - \sigma_u\) for all Dirichlet series. The Bohr–Bohnenblust–Hille theorem states that \(T=\frac12\). The authors’ main aim is to extend the Bohr–Bohnenblust–Hille theorem from \(H_\infty\big(B_{\ell_\infty}\big)\) to any set \(\mathcal F\big(\ell_w \cap \mathbb D^{\mathbb N}\big)\) of holomorphic functions satisfying property (T):
Given \(1\leq w\leq\infty\), a set of holomorphic functions \(\mathcal F\big(\ell_w \cap \mathbb D^{\mathbb N}\big)\) has property (T), if it remains stable under composition with diagonal operators of norm \(\leq 1\) [we do not repeat the definition of “stable”, which is given at the beginning of §2], and if it contains the continuous polynomials on \(\ell_w\) as well as all bounded holomorphic functions on \(\ell_w \cap \mathbb D^{\mathbb N}\).
The authors’ main result is (Theorem 3.2): Given \(1\leq w\leq \infty\) and a set \(\mathcal F\big(\ell_w \cap \mathbb D^{\mathbb N}\big)\) of holomorphic functions containing the polynomials on \(\ell_w\), then
\[ S\Big(\mathcal F\big(\ell_w \cap \mathbb D^{\mathbb N}\big)\Big) = \begin{cases} 1 & \text{if \(1\leq w<2\)}, \\ \left(\frac12 +\frac1w\right)^{-1} & \text{if \( 2\leq w \leq \infty\)},\end{cases} \]
and if, in addition, \(\mathcal F\big(\ell_w \cap \mathbb D^{\mathbb N}\big)\) satisfies property (T), then \[ T\Big(\mathcal F\big(\ell_w \cap \mathbb D^{\mathbb N}\big)\Big) = \frac1{S\Big(\mathcal F\big(\ell_w \cap \mathbb D^{\mathbb N}\big)\Big)} = \begin{cases} 1 & \text{if \(1\leq w<2\)}, \\ \left(\frac12 +\frac1w\right) & \text{if \( 2\leq w \leq \infty\)}.\end{cases} \]
So, the Bohr–Bohnenblust–Hille theorem can be extended from \(H_\infty(B_{\ell_\infty})\) to any set \(\mathcal F\big(\ell_w \cap \mathbb D^{\mathbb N}\big)\) of holomorphic functions satisfying property (T).
Here \(\mathbb D\) is the open unit disc in \(\mathbb C\), \(\mathcal F(R)\) is a set of holomorphic functions \(f\) on \(R\), a Reinhardt domain in \(\ell_w\), \(f\) has a formal power series expansion \(\sum_{\alpha \in \mathbb N_0^{(\mathbb N)}} c_\alpha \cdot z^\alpha\), \(\text{dom\,}\mathcal F(R)\), the domain of convergence of \(\mathcal F(R)\), is
\[ \text{dom}\,\mathcal F(R) = \Bigg\{ z\in R:\; \sum_{\alpha \in \mathbb N_0^{(\mathbb N)}} | c_\alpha(f) z^\alpha | <\infty, \text{ for all }f\in \mathcal F(R)\Bigg\}. \]
For a Reinhardt domain \(R\) in \(\ell_w\) (\(1\leq w\leq\infty\)) and a set \(\mathcal F(R)\) of holomorphic functions on \(R\), one defines
\[ S\big(\mathcal F(R)\big) = \sup \Big\{q\geq 1: \ell_q \cap R \subset \text{dom} \mathcal F(R)\Big\}, \] and
\[ T\big(\mathcal F(R)\big) = \sup \Big\{ \sigma_\alpha - h(\mathcal F(R))\Big\}, \]
where this supremum is taken over all Dirichlet series. Here
\[ h\big(\mathcal F(R)\big) = \inf\Bigg\{ \sigma \in \mathbb R:\; g_\sigma(z) = \sum_{\alpha \in \mathbb N_0^{(\mathbb N)}} \frac{a_{p^\alpha}}{p^{\alpha \,\sigma}} \; z^\alpha \in \mathcal F(R)\Bigg\}. \]
Proposition 3.1 shows that in the special case \(\mathcal F(R) = H_\infty \big(\ell_w \cap \mathbb D^{\mathbb N}\big)\), these definitions are the same as Bohr’s definitions of \(S\) and \(T\).

Reviewer: Wolfgang Schwarz (Frankfurt am Main)

MSC:

30B50	Dirichlet series, exponential series and other series in one complex variable
32A05	Power series, series of functions of several complex variables
32A10	Holomorphic functions of several complex variables

Keywords:

abscissa of absolute convergence; abscissa of uniform convergence; Dirichlet series; formal power series with infinitely many variables; Reinhardt domains; infinite dimensional holomorphy