Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables

Abstract

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 ${\overline{\lim }}_n\frac{1}{\log n}{\sum }_{j=1}^n{b}_{p_j}^{\ast 2}<1$, and conversely, if $b$ is an ${\ell }_1$-multiplier for ${\mathcal{H}}_{\infty }$, then $|b_{p_j}|<1$ for all $j$ and ${\overline{\lim }}_n\frac{1}{\log n}{\sum }_{j=1}^n{b}_{p_j}^{\ast 2}\le 1$ (here $b^{\ast }$ 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 ${\mathbb{T}}^{\infty }$.

arXiv admin note: substantial text overlap with arXiv:1207.2248

References in corpus (2)

• The Bohnenblust--Hille inequality for homogeneous polynomials is hypercontractive
• Fatou and brother Riesz theorems in the infinite-dimensional polydisc

Cited by in corpus (1)

• Monomial convergence on ${\ell }_r$