Multilinearity in one and two dimensions

该提案描述了旨在回答遍历理论，谐波分析和算术组合中的一系列问题的研究计划，这些问题在现代分析中非常感兴趣。所有这些问题的主要共同特征在于它们的多线性性质。当处理作用于勒贝格空间的积上的多线性算子时，特别有趣的是理解它表现良好的指标范围。在这个建议中，重点主要是在几乎处处收敛，从而固有的最大算子的有界性。一个著名的未解决的问题是关于与交换变换相关的双线性遍历平均的点向收敛性。对这个问题的分析揭示了它与二维双线性希尔伯特变换和Carleson关于傅立叶级数点向收敛的定理之间的深刻联系。另一个有趣的问题出现在双线性多项式平均的分析中。结合多尺度时频技术的算术组合学的最新进展可能会揭示所有这些问题。这项提议的研究是目前在动力系统、谐波分析和算术组合学中所做的研究的前沿。我们期望，本提案中提出的问题的解决将进一步促进数学界对这些领域之间联系的理解，特别是谐波分析过程与遍历理论中类似过程之间的联系。这项研究的性质也使我们的调查更有可能揭示一些在其他科学领域使用的工具，比如信号处理。这项拨款所提出的研究项目将导致PI与其他大学的数学家之间的互动，他们可能会进行部分调查。

This proposal describes research plans that are aimed at answering an array of questions in ergodic theory, harmonic analysis and arithmetic combinatorics that are of great interest in modern analysis. The main common feature of all these problems resides in their multilinear nature. Of particular interest when dealing with a multilinear 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 associated maximal operators. A famous unresolved problem regards pointwise convergence for the bilinear ergodic averages associated with commuting transformations. The analysis of this question revealed deep connections with a two dimensional version of the bilinear Hilbert transform and with Carleson's theorem on the pointwise convergence of Fourier series. Another interesting circle of questions arises in the analysis of bilinear polynomial averages. Recent progress from arithmetic combinatorics, combined with multi-scale time-frequency techniques is likely to shed light on all these issues.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 analogues 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.