Mathematical Sciences: Singular Integrals and Fourier Integral Operators

数学科学：奇异积分和傅里叶积分算子

• 批准号: 9204196
• 负责人: Duong Phong
• 金额: $ 14.5万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-05-01 至 1995-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204196&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Singular; Integrals; Fourier

This project seeks to analyze the mathematical theories of singular integral, Fourier integral and pseudodifferential operators. The theories developed over the past thirty years yielding a wide range of applications based on their simplicity and unity. Nevertheless, operators falling outside their scope are appearing with increased frequency in studies of Fourier-Airy operators of diffraction theory, maximal operators and Hilbert transforms along curves of differentiation theory, Hilbert integral operators and singular Radon transforms of subelliptic boundary value problems, oscillatory singular integrals on nilpotent groups, the X-ray transforms and Guillemin operators of integral geometry and deformation theory on manifolds of geodesics. Two unifying themes have emerged; they appear themselves to be related. The first consists of Fourier integral operators whose Lagrangian is not a local graph but projects on tangent spaces with folds or cusps. The second is where the densities on the Lagrangian have singularities. The main goal of this work is to arrive at a more complete theory of degenerate Fourier integral and singular integral operators. Strong evidence for the existence of such a framework is provided by recent progress in several directions: work will be done in exploring them further. First, efforts will be made to understand the stratification of the singular varieties of projections of degenerate operators on tangent spaces and how they influence Sobolev bounds. This will only treat operators with constant coefficients. Efforts at a formulation of the theory on manifolds, a new phenomenon - not present in the classical case - occurs. Work must be done in obtaining lower bounds for the oscillation of the phase function. Other goals involve finding extensions to operators defined in spaces of higher dimension and the measure of the level of stratification of the local geometry of the Lagrangian to yield sharp bounds for singular Radon transforms.

该项目旨在分析单数积分，傅立叶积分和伪差异操作员的数学理论。 在过去的三十年中，这些理论根据其简单性和统一性产生了广泛的应用。 Nevertheless, operators falling outside their scope are appearing with increased frequency in studies of Fourier-Airy operators of diffraction theory, maximal operators and Hilbert transforms along curves of differentiation theory, Hilbert integral operators and singular Radon transforms of subelliptic boundary value problems, oscillatory singular integrals on nilpotent groups, the X-ray transforms and Guillemin operators of integral geometry and deformation theory on地球学的流形。 出现了两个统一的主题；他们似乎是相关的。 第一个由傅立叶积分运算符组成，其拉格朗日不是本地图，而是带有折叠或尖端的切线空间上的项目。 第二个是拉格朗日人的密度具有奇异性的地方。 这项工作的主要目标是提出更完整的傅里叶积分和奇异整体运营商的理论。 在多个方向上的最新进展提供了有力的存在这种框架的证据：将在进一步探索它们方面完成工作。 首先，将做出努力，以了解归化运算符的奇异变体在切线空间上的分层以及它们如何影响Sobolev的界限。 这只会以恒定的系数处理操作员。 为了形成多种理论的努力，发生了一种新现象 - 在经典案例中不存在。 必须在获得相位函数振荡的下限时完成工作。 其他目标涉及在较高尺寸的空间和拉格朗日局部几何分层水平的量度中找到对操作员的扩展，以产生奇异ra尺的尖锐界限。