New strips of convergence for Dirichlet series
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 \[ \begin{matrix} \sigma_a\\ \sigma_u\end{matrix} = \inf_{r\in\mathbb R} \left\{ \sum_1^\infty \frac{a_n}{n^s}\quad\begin{matrix}\text{absolutely}\\ \text{uniformly}\end{matrix}\text{ convergent on }\Re(s) \geq r\right\}. \] 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 \S\ 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 \(T\).
formal power series with infinitely many variables
scientific article; zbMATH DE number 5770005