Mathematical Sciences: Singular Integrals and Fourier Integrals

摘要Greenleaf 9531806 我们将把以前得到的波动方程解的特征时空估计推广应用于从后向散射的近似知识（或其他确定的散射数据集）中近似确定三维空间L^2中紧支撑的与时间无关的势的结果。扩展这样的结果与非紧的支持和N-粒子类型的潜力将继续下去。估计，在索伯列夫和勒贝格规范，傅立叶积分算子与典型的关系表现出尖点奇点，无论是简单的和高阶的，将被调查。这样的估计有应用程序的正则性平均运营商与一般家庭的直线和曲线在n维空间中的积分。最后，研究了n维射影空间中最小次数代数变量的超平面截面的Goncharov复形的真实的类似物所产生的一类Fourier积分算子，目的是得到它们的合成演算。 本计画的主要目的是研究几种奇异积分算子与傅立叶积分算子。这种算子将一个空间上的函数转换为另一个（可能不同）空间上的函数，已经成为研究线性偏微分方程的核心工具，这些方程支配着各种物理现象，如电磁场和声音传播。在这个项目中要研究的特定运营商出现在散射波的潜在功能，并在断层扫描，数学基础上的各种医疗成像系统，如CAT和MRI扫描仪。因此，在这个项目中考虑的问题的进展将有助于重建未知量的物理感兴趣的非侵入性的观察程序的理论基础。尽管事实上，他们出现在不同的问题，要研究的运营商有几个共同的特点，这涉及到更复杂的几何比目前在原始版本的奇异积分和傅立叶积分算子。我们希望，最终，这些运营商的理解将导致改善重建技术。

Abstract Greenleaf 9531806 Characteristic space-time estimates, previously obtained, for solutions of wave equations will be extended and applied to yield results on the approximate determination of compactly supported, time-independent potentials in L^2 of 3-space from approximate knowledge of their backscattering (or other determined sets of scattering data.) Extensions of such results to potentials with noncompact support and of N-particle type will be pursued. Estimates, in terms of Sobolev and Lebesgue norms, for Fourier integral operators associated with canonical relations exhibiting cusp singularities, both simple and of higher order, will be investigated. Such estimates have applications to regularity properties of averaging operators associated with integrals over generic families of lines and curves in n-dimensional space. Finally, classes of Fourier integral operators arising from real analogues of Goncharov's complexes of hyperplane sections of algebraic arieties of minimal degree in n-dimensional projective space will be studied, with the goal of obtaining composition calculi for them. The principal object of this project will be the study of several types of singular integral operators and Fourier integral operators. Such operators, which transform functions on one space into functions on another (possibly different) space, have become central tools in the study of linear partial differential equations which govern diverse physical phenomena, such as electromagnetic fields and sound propagation. The particular operators to be studied in this project arise in the scattering of waves by potential functions, and in tomography, the mathematical basis for a variety of medical imaging systems, such as CAT and MRI scanners. Thus, progress on the problems considered in this project will contribute to the theoretical underpinnings of procedures for reconstructing unknown quantities of physical interest from noninvasive observations. Despite the fact that they arise in d ifferent problems, the operators to be studied share several common features, which involve more complicated geometry than is present in the original versions of singular integral and Fourier integral operators. It is hoped that, eventually, improved understanding of these operators will lead to improved reconstruction techniques.