Mathematical Sciences: Singular Integrals and Fourier Integral Operators

本项目致力于分析奇异积分、傅里叶积分和拟微分算子的数学理论。这些理论在过去的三十年里发展起来，以其简单性和统一性为基础，产生了广泛的应用。然而，在绕射理论的傅里叶-艾里算子、沿微分理论曲线的极大算子和希尔伯特变换、次椭圆边值问题的希尔伯特积分算子和奇异Radon变换、幂零群上的振荡奇异积分、测地线流形上积分几何和形变理论的X射线变换和Guillmin算子。出现了两个统一的主题；它们似乎相互关联。第一类是傅立叶积分算子，它的拉格朗日不是局部图，而是投影在具有折线或尖点的切线空间上。第二个是拉格朗日上的密度有奇点的地方。这项工作的主要目的是得到一个更完整的退化傅立叶积分和奇异积分算子的理论。最近在几个方向上取得的进展有力地证明了这一框架的存在：将在进一步探讨这些问题方面开展工作。首先，将努力了解退化算子在切线空间上的奇异投影种类的分层以及它们是如何影响Sobolev界的。这将只处理具有常系数的运算符。关于流形的理论的公式化的努力，一种新的现象--在经典情况下不存在--发生了。必须做的工作是获得相函数振荡的下界。其他目标包括找到定义在更高维空间中的算子的扩展，以及拉格朗日局部几何的分层程度的度量，以产生奇异Radon变换的锐界。

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.