Bilinear techniques in time-frequency and real analysis

这一建议将促进研究者在傅里叶分析和实态分析领域的研究。所提出的问题是双线性伪微分算子和加权双线性poincar<e:1>和Sobolev不等式理论的核心，包括双线性技术在时频和实分析中最基本水平的发展和实现，从而扩大了它们在分析和偏微分方程中的应用范围。为了进一步达到这些目的，我们特别注意研究双线性伪微分算子和分子副积在Lebesgue和调制空间中的有界性，并通过研究carnot - carath<s:1>空间中的双线性表示公式和双线性分数积分算子，研究加权双线性poincar<e:1>和Sobolev不等式。本研究计划考虑的其他相关功能空间包括Sobolev、Besov、triiebel - lizorkin空间、BMO、弱Lebesgue空间和Campanato-Morrey空间的尺度及其加权版本。傅立叶分析自诞生以来，在数学、物理和工程的各个领域做出了许多重大贡献；通过该计划开展的研究将积极地继续增加这些学科。提出的研究特别适用于流体动力学、量子力学和光学等物理领域的非线性偏微分方程理论。该项目还将促进博士后、研究生和本科水平的研究与教育的整合，促进发现、形成人力资源和开发学术课程。

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.