Multilinearity in one and two dimensions

一维和二维的多重线性

• 批准号: 0901208
• 负责人: Ciprian Demeter
• 金额: $ 14.77万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901208&HistoricalAwards=false
• 关键词: Multilinearity; two; dimensions

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.

这项建议描述了旨在回答现代分析中非常感兴趣的遍历理论、调和分析和算术组合学中的一系列问题的研究计划。所有这些问题的主要共同特征在于它们的多线性性质。当处理作用在勒贝格空间的某些乘积上的多线性算子时，特别感兴趣的是了解它对其表现良好的指数的范围。在这个方案中，焦点主要集中在几乎处处收敛，从而内在地关注相关极大算子的有界性。一个著名的悬而未决的问题是关于与交换变换相关的双线性遍历平均的逐点收敛。对这个问题的分析揭示了与二维双线性希尔伯特变换和Carleson关于傅里叶级数逐点收敛的定理的深刻联系。另一个有趣的问题出现在双线性多项式平均的分析中。算术组合学的最新进展，结合多尺度时频技术，可能会揭示所有这些问题。这项拟议的研究处于动力系统、调和分析和算术组合学的前沿。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其遍历理论中的类似过程之间的联系。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。