Mathematical Sciences: Singular Integrals and Fourier Integral Operators

该项目旨在分析的数学理论， 奇异积分、Fourier积分和伪微分 运营商 这些理论是在过去三十年中发展起来的 基于其简单性产生了广泛的应用 和团结 然而，不属于其范围的经营者 在傅立叶-艾里的研究中出现的频率越来越高 衍射理论的算子，极大算子和希尔伯特 微分理论中的沿着曲线变换 积分算子与次椭圆奇异Radon变换 边值问题，振荡奇异积分， 幂零群，X射线变换和Guillemin算子 流形上积分几何和形变理论 测地线 出现了两个统一的主题;它们似乎 有关系。 第一类由傅里叶积分算子组成 其拉格朗日量不是局部图而是投影在切线上 有褶皱或者尖点的空间。 第二个是， 拉格朗日函数有奇点。 这项工作的主要目标是达到一个更完整的 退化Fourier积分与奇异积分理论 运营商 强有力的证据表明，存在这样一个框架， 最近在几个方面取得了进展：将开展工作， 进一步探索它们。 一是着力 了解奇异品种的分层 退化算子在切空间上的投影以及如何 它们影响Sobolev边界。 这将只对待运营商 常数系数。 努力拟订一项 理论上的流形，一个新的现象-不存在于 经典案例-发生。 必须努力降低 相位函数振荡的边界。 其他目标 涉及到寻找定义在空间中的算子的扩展， 更高的维度和分层水平的度量 拉格朗日量的局部几何产生尖锐的界限， 奇异Radon变换

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.