Dirichlet series and complex analysis for functions in infinitely many variables

近年来，我们已经看到了显着增长的兴趣，在某些功能分析方面的理论，普通狄利克雷级数$\sum_n a_n n^{-s}$。它变得越来越清楚，理论的自然类的普通狄利克雷级数通过玻尔的基本工作是密切相关的无限维全纯和傅立叶分析的职能在无穷多个变量。哈代空间$\mathcal{H}_p$给出了最重要和最深入研究的普通狄利克雷级数类。当代的研究似乎几乎只涉及普通的狄利克雷级数，尽管理论的创始人如玻尔、朗道、哈代和里兹（以及其他人）从所谓的一般狄利克雷级数$\sum_n a_n e^{-\lambda_n s}$开始深入研究，其中$\lambda=（\lambda_n）$是一个严格递增的正真实的数序列，趋向于$\infty$（称为频率）。我们的项目的主要目标是再现一般Dirichlet空间（固定频率）的$\mathcal{H}_p$-理论的普通理论的一些关键结果。这似乎是一项艰巨的任务（至少对某些主题），因为设置一般狄利克雷级数当然是更广泛的比普通的。 事实上，最大的挑战是找到合理的替代品的戏剧性损失的工具，从复杂的分析和傅立叶分析的无限维多圆盘和环面。

In recent years we have seen a remarkable growth of interest in certain functional analytic aspects of the theory of ordinary Dirichlet series $\sum_n a_n n^{-s}$. It became more and more clear that the theory of natural classes of ordinary Dirichlet series through Bohr's fundamental work is intimately connected with infinite dimensional holomorphy and Fourier analysis for functions in infinitely many variables. The most important and most intensively studied classes of ordinary Dirichlet series are given by the Hardy spaces $\mathcal{H}_p$. Contemporary research almost exclusively seems to deal with ordinary Dirichlet series, although the founding fathers of the theory like Bohr, Landau, Hardy and Riesz (among others) started with deep work on so-called general Dirichlet series $\sum_n a_n e^{-\lambda_n s}$ where $\lambda= (\lambda_n)$ is a strictly increasing sequence of positive real numbers tending to $\infty$ (called frequency). The main goal of our project is to reproduce some of the key results of the ordinary theory of $\mathcal{H}_p$-theory for general Dirichlet spaces (fixing a frequency). This seems a difficult task (at least for some topics) since the setting of general Dirichlet series is of course much broader than that of ordinary ones. In fact, the big challenge is to find reasonable replacements for the dramatic loss of tools from complex analysis and Fourier analysis on the infinite dimensional poly disc and torus.