Dirichlet series and complex analysis for functions in infinitely many variables

近年来，人们对普通狄里克莱级数理论的某些泛函分析方面的兴趣有了显着的增长。通过玻尔的基本工作，人们越来越清楚地认识到，普通Dirichlet级数的自然类理论与无穷维全纯和无穷多元函数的傅里叶分析密切相关。一般Dirichlet级数中最重要和最受研究的类是由Hardy空间{H}_p给出的。当代的研究似乎只涉及普通的狄里克莱级数，尽管这一理论的创始人如玻尔、兰道、哈代和里兹(以及其他人)开始于对所谓的广义狄里克莱级数$\sum_n a_n e^{-\lambda_n S}$的深入研究，其中$\lambda=(\lambda_n)$是趋近于$\inty$(称为频率)的严格递增的正实数序列。我们项目的主要目的是重现一般Dirichlet空间(固定频率)的一般理论的一些关键结果。这似乎是一项艰巨的任务(至少对某些主题而言)，因为一般狄利克莱级数的背景当然比普通系列的背景要宽泛得多。事实上，最大的挑战是找到合理的替代方法，以弥补在无限维多维圆盘和环面上进行复分析和傅立叶分析所造成的巨大工具损失。

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.