Monomial expansions of $H_{p}$--functions in infinitely many variables

Abstract

Each bounded holomorphic function on the infinite dimensional polydisk $\mathbb{D}^\infty$, $f \in H_\infty(\mathbb{D}^\infty)$, defines a formal monomial series expansion that in general does not converge to $f$. The set $\mon H_\infty(\mathbb{D}^\infty)$ contains all $ z $'s in which the monomial series expansion of each function $f \in H_\infty(\mathbb{D}^\infty)$ sums up to $f(z)$. Bohr, Bohnenblust and Hille, showed that it contains $\ell_{2} \cap \mathbb{D}^\infty$, but does not contain any of the slices $\ell_{2+\varepsilon} \cap \mathbb{D}^\infty$. This was done in the context of Dirichlet series and our article is very much inspired by recent deep developments in this direction. Our main contribution shows that $z \in \mon H_\infty(\mathbb{D}^\infty)$ whenever $\bar{\lim} \big(\frac{1}{\log n} \sum_{j=1}^{n} z^{* 2}_{j} \big)^{1/2} < 1/\sqrt{2}$, and conversely $\bar{\lim} \big(\frac{1}{\log n} \sum_{j=1}^{n} z^{* 2}_{j} \big)^{1/2} \leq 1$ for each $z \in \mon H_\infty(\mathbb{D}^\infty)$. The Banach space $H_\infty(\mathbb{D}^\infty)$ can be identified with the Hardy space $H_\infty(\mathbb{T}^\infty)$; this motivates a study of sets of monomial convergence of $H_p$-functions on $\mathbb{T}^\infty$ (consisting of all $z$'s in $\mathbb{D}^{\infty}$ for which the series $\sum \hat{f}(\alpha) z^{\alpha}$ converges). We show that $\mon H_\infty(\mathbb{T}^\infty) = \mon H_\infty(\mathbb{D}^\infty)$ and $\mon H_{p}(\mathbb{T}^\infty) = \ell_{2} \cap \mathbb{D}^\infty$ for $1 \leq p < \infty$ and give a representation of $H_{p}(\mathbb{T}^\infty)$ in terms of holomorphic functions on $\mathbb{D}^{\infty}$. This links our circle of ideas with well-known results due to Cole and Gamelin.