$H_{p}$的单式展开--无穷多变量函数
Monomial expansions of $H_{p}$--functions in infinitely many variables
摘要：
无限维多圆盘$\mathbb{D}^\infty$上的每一个有界全纯函数，$f \in H_\infty(\mathbb{D}^\infty)$，定义了一个形式的单项级数展开式，该展开式一般不收敛于$f$。集合$\mon H_\infty(\mathbb{D}^\infty)$包含所有$ z $，其中每个函数$f \in H_\infty(\mathbb{D}^\infty)$的单项级数展开求和为$f(z)$。Bohr、Bohnenblust和Hille表明，它含有$\ell_{2} \cap \mathbb{D}^\infty$，但不含有任何$\ell_{2+\varepsilon} \cap \mathbb{D}^\infty$切片。这是在狄利克雷级数的背景下完成的，我们的文章受到最近在这个方向上的深入发展的启发。我们的主要贡献表明，$z \in \mon H_\infty(\mathbb{D}^\infty)$每当$\bar{\lim} \big(\frac{1}{\log n} \sum_{j=1}^{n} z^{* 2}_{j} \big)^{1/2} < 1/\sqrt{2}$，相反，$\bar{\lim} \big(\frac{1}{\log n} \sum_{j=1}^{n} z^{* 2}_{j} \big)^{1/2} \leq 1$为每个$z \in \mon H_\infty(\mathbb{D}^\infty)$。巴拿赫空间$H_\infty(\mathbb{D}^\infty)$可以确定与哈代空间$H_\infty(\mathbb{T}^\infty)$ ;这激发了研究集单项收敛的$H_p$ -功能$\mathbb{T}^\infty$（包括所有$z$的在$\mathbb{D}^{\infty}$的一系列$\sum \hat{f}(α) z^α$收敛）。证明了$1 \leq p < \infty$的$\mon H_\infty(\mathbb{T}^\infty) = \mon H_\infty(\mathbb{D}^\infty)$和$\mon H_{p}(\mathbb{T}^\infty) = \ell_{2} \cap \mathbb{D}^\infty$，并给出了$H_{p}(\mathbb{T}^\infty)$在$\mathbb{D}^{\infty}$上的全纯函数表示.这将我们的思想圈与科尔和甘默林的著名结果联系起来。
英文摘要：
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 $\sum \hat{f}(α) z^α$ 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.
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