Mathematical Sciences: Singular Integrals and Fourier Integrals

先前得到的波动方程解的特征时空估计将得到扩展，并应用于从它们的后向散射（或其他确定的散射数据集）的近似知识近似确定三维空间L^2中紧支持的、时间无关的势的结果。将这些结果推广到具有非紧致支持和n粒子型的势。估计，在Sobolev和Lebesgue范数，与典型关系相关的傅里叶积分算子显示尖点奇点，简单和高阶，将进行研究。这种估计可以应用于n维空间中与一般直线和曲线族上的积分相关的平均算子的正则性。最后，研究了n维射影空间中最小次代数变元的超平面截面Goncharov复形的实类似物所产生的傅里叶积分算子类，目的是获得它们的复合演算。这个项目的主要目标是研究几种类型的奇异积分算子和傅里叶积分算子。这些算子将一个空间上的函数转换为另一个（可能不同的）空间上的函数，已经成为研究线性偏微分方程的核心工具，这些方程控制着各种物理现象，如电磁场和声音传播。本项目中要研究的特定操作符出现在势函数对波的散射和断层扫描中，断层扫描是各种医学成像系统（如CAT和MRI扫描仪）的数学基础。因此，本项目中所考虑的问题的进展将有助于从非侵入性观测中重建未知数量的物理兴趣的理论基础。尽管它们出现在不同的问题中，但要研究的算子有几个共同的特征，它们涉及比原始版本的奇异积分和傅里叶积分算子更复杂的几何。人们希望，最终，对这些操作人员的进一步了解将导致重建技术的改进。

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.