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 ${\ell }_{2+\epsilon }\cap {\mathbb{D}}^{\infty }$. 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 $\bar{\lim }(\frac{1}{\log n}{\sum }_{j=1}^n{z}_j^{\ast 2}{)}^{1/2}<1/\sqrt{2}$, and conversely $\bar{\lim }(\frac{1}{\log n}{\sum }_{j=1}^n{z}_j^{\ast 2}{)}^{1/2}\le 1$ 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 $1\le p<\infty$ 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.