Mathematical Sciences: Singular Integrals and Fourier Integrals

数学科学：奇异积分和傅立叶积分

• 批准号: 9101298
• 负责人: Allan Greenleaf
• 金额: $ 2.4万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-07-15 至 1993-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9101298&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Singular; Integrals; Fourier

In this project the principal investigator will study several problems associated with singular integrals and Fourier integrals. In particular, he will formulate classes of canonical relations that apply to examples from integral geometry and isospectral deformations, and he will prove composition calculi and sharp L-2 estimates for Fourier integral operators associated with them. The principal investigator will also look into the local solvability of systems of first-order pseudodifferential operators. Finally he will obtain sharper estimates for degenerate pseudodifferential operators with singular symbols, and he will apply these results to degenerate Radon transforms. A number of problems in theoretical and applied analysis concern the inversion of an operation or a transformation. For example, the desired solution of a problem may be transformed into an equation that involves integrals or integral operators, and then in order to obtain the solution, one must invert the transformation, that is, solve the integral equation. In this project the principal investigator will analyze two classes of integral operators, in order to provide sharp estimates for the solutions they represent. One application of this work is to the estimation of solutions of X-ray tomography problems.

在这个项目中，首席研究员将研究与奇异积分和傅立叶积分相关的几个问题。特别地，他将阐述适用于积分几何和等谱变形例子的正则关系的类别，他将证明与之相关的傅立叶积分算子的组成演算和尖锐的L-2估计。主要研究者还将研究一阶伪微分算子系统的局部可解性。最后，他将获得具有奇异符号的退化伪微分算子的更清晰的估计，并将这些结果应用于退化Radon变换。理论分析和应用分析中的许多问题都与运算或变换的反转有关。例如，一个问题的期望解可能被转换成一个包含积分或积分算子的方程，然后为了得到解，必须进行逆变换，即求解积分方程。在这个项目中，主要研究者将分析两类积分算子，以便为它们所代表的解提供清晰的估计。这项工作的一个应用是对x射线断层成像问题的解的估计。