Bilinear techniques in time-frequency and real analysis

时频和实分析中的双线性技术

• 批准号: 1101327
• 负责人: Virginia Naibo
• 金额: $ 11.75万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101327&HistoricalAwards=false
• 关键词: Bilinear; techniques; time; frequency; real

This proposal will advance the investigator's research in the areas of Fourier and real analysis. The posed problems lie at the core of the theories of bilinear pseudodifferential operators and weighted bilinear Poincaré and Sobolev inequalities and include the development and implementation of bilinear techniques at their most fundamental level in time-frequency and real analysis, thus broadening the scope of their applications to Analysis and Partial Differential Equations. To further these ends, particular attention is given to the study of boundedness properties in the setting of Lebesgue and modulation spaces of bilinear pseudodifferential operators and molecular paraproducts, and to weighted bilinear Poincaré and Sobolev inequalities through the study of bilinear representation formulas and bilinear fractional integral operators in the context of Carnot-Carathéodory spaces. Other relevant function spaces considered in this research program include the scales of Sobolev, Besov, Triebel-Lizorkin spaces, BMO, weak Lebesgue spaces, and Campanato-Morrey spaces as well as their weighted versions.Fourier Analysis has since its origins made many significant contributions to various areas of mathematics, physics and engineering; the research developed through this program will positively continue to add to these disciplines. The proposed research has in particular applications to the theory of nonlinear Partial Differential Equations from areas of physics such as fluid dynamics, quantum mechanics and optics. This project will also contribute to the integration of research and education at the postdoctoral, graduate, and undergraduate levels, to advancing discovery, forming human resources, and developing academic curriculum.

这一建议将推动调查员在傅立叶和真实的分析领域的研究。所提出的问题在于双线性伪微分算子和加权双线性庞加莱和Sobolev不等式的理论的核心，包括发展和实施的双线性技术在其最基本的水平，在时间-频率和真实的分析，从而扩大了其应用范围的分析和偏微分方程。为了进一步实现这些目的，特别注意研究双线性伪微分算子和分子仿积在勒贝格和调制空间中的有界性，并通过研究Carnot-Carathéodory空间中的双线性表示公式和双线性分数次积分算子来研究加权双线性Poincaré和Sobolev不等式。其他相关的函数空间包括Sobolev，Besov，Triebel-Lizorkin空间，BMO，弱Lebesgue空间和Campanato-Morrey空间及其加权形式。傅立叶分析自其起源以来，对数学，物理和工程的各个领域做出了许多重要贡献;通过这个计划开发的研究将积极地继续增加这些学科。拟议的研究特别适用于流体力学、量子力学和光学等物理领域的非线性偏微分方程理论。该项目还将有助于整合博士后，研究生和本科生水平的研究和教育，以推进发现，形成人力资源和开发学术课程。