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.

该提案描述了研究计划，旨在回答遍历理论，调和分析和算术组合学中的一系列问题，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点在于其多线性。 当处理作用于勒贝格空间的某些乘积的多线性算子时，特别感兴趣的是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，从而内在的相关极大算子的有界性。一个著名的未解决的问题，关于点态收敛的双线性遍历平均与交换变换。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关于傅立叶级数逐点收敛定理的深层联系。 另一个有趣的循环问题出现在双线性多项式平均值的分析中。算术组合学与多尺度时频技术相结合的最新进展可能会阐明所有这些问题。这一拟议中的研究处于动力系统、谐波分析和算术组合学领域的前沿。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遍历理论中的类似物之间的过程。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。