Multilinear Operators in Harmonic Analysis and Ergodic Theory

本提案描述了旨在回答遍历理论和调和分析中的一系列问题的研究计划，这些问题是现代分析中非常感兴趣的。所有这些问题的主要共同特征在于它们的多线性性质。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，理解它表现良好的指数范围。在这个方案中，焦点主要集中在几乎处处收敛，因此本质上是关于结合的极大算子的有界性。多重线性的一个重要例子是组合平均，如Furstenberg的非标准平均和立方体上的平均。这些平均的收敛对于整数上的各种Ramsey型问题甚至一般抽象群上的各种Ramsey型问题都有深刻的意义。我们计划研究的第二类多线性算子是由加权平均和级数表示的，例如出现在所谓的“返回时间定理”中的那些算子。对这些对象的分析揭示了动力学和时频分析之间的显著联系，特别是Birghoff逐点遍历定理和Carleson关于傅立叶级数逐点收敛的结果之间的联系。这项拟议的研究是目前在动力系统、调和分析和算术组合学中所做的前沿研究。预计本提案中提出的问题的解决将促进数学界对这些领域之间的联系的理解，特别是调和分析中的过程与它们在遍历理论中的类比之间的联系。这项研究的性质也使我们的调查有可能进一步阐明其他科学领域中使用的一些工具，如信号处理。这笔赠款中提议的研究项目将导致PI与其他大学的数学家之间的互动，这些大学可能会与其他大学进行部分调查。

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.