Multilinear Operators in Harmonic Analysis and Ergodic Theory

摘要该建议描述了研究计划，旨在回答一系列问题，遍历理论和谐波分析是非常感兴趣的现代分析。 所有这些问题的主要共同特点在于其多线性。特别有趣的是，当处理作用于勒贝格空间的某些乘积的多线性算子时，要理解它表现良好的指数范围。在这个建议的重点主要是几乎处处收敛，从而内在的associatedmaximum运营商的有界性。多重线性的一个重要例子是组合平均数，如Furstenberg的非标准平均数和立方体上的平均数。这种平均值的收敛性对整数甚至一般抽象群上的各种拉姆齐型问题都有深刻的影响。我们计划研究的第二种类型的多线性算子由加权平均和级数表示，例如出现在所谓的“返回时间定理”中的算子。对这些物体的分析揭示了动力学和时频分析之间的惊人联系，特别是Birghoff的逐点遍历定理和Carleson的Fourier级数的逐点收敛结果之间的联系。 这项研究是目前在动力系统，谐波分析和算术组合学正在做的最前沿。预计，在这个建议中提出的问题的解决将进一步数学界的理解这些领域之间的联系，特别是在调和分析和它们的类似物遍历理论的过程之间。这项研究的性质也使我们的调查有可能为其他科学领域（如信号处理）中使用的一些工具提供更多的信息。这项拨款中提出的研究项目将导致PI和其他大学的数学家之间的互动，其中部分调查可能会进行。

ABSTRACTThis proposal describes research plans that are aimed at answering an array of questions in ergodic theory and harmonic analysis that are of great interest in modern analysis. The main common feature of all these problems resides in their multi-linear nature. Of particularInterest, when dealing with a multi-linear operator acting on some product of Lebesgue spaces, is to understand the range of indices for which it is well behaved. In this proposal the focus is mainly on almost everywhere convergence, and thus inherently on the boundedness of the associatedmaximal operators. An important instance of multi-linearity is represented by the combinatorial averages such as Furstenberg's nonstandard averages and the averages on cubes. The convergence of such averages has deep implications for various Ramsey-type problems on integers and even on general abstract groups. A second type of multi-linear operators we plan to investigate is represented by the weighted averages and series, such as the ones appearing in the so-called ``return times theorems''. The analysis of these objects reveals striking connections between dynamics and time-frequency analysis, in particular between Birghoff's point-wise ergodic theorem and Carleson's result on the point-wise convergence of Fourier series. This proposed research is at the cutting edge of what is now being done in dynamical systems, harmonic analysis and arithmetic combinatorics. It is expected that the resolution of the questions advanced in this proposal will further the mathematical community's understanding of the connections between these areas, in particular between processes in harmonic analysis and their analogue in ergodic theory. The nature of this research makes it also likely for our investigation to shed yet more light on some of the tools that are used in other areas of science, such as signal processing. The research project that is proposed in this grant will lead to interactions between the PI and mathematicians from other universities with whom part of the investigation might be conducted.