Fourier Integrals and Generalized Radon Transforms

摘要：本文将研究与正则关系相关的傅里叶积分算子的估计，其投影具有稳定的奇点类，特别是高阶尖点。对于双面尖头，将发展奇异品种附近算子的新分解。这些估计和分解将应用于广义Radon变换的有界性，特别是Sobolev和Lebesgue空间上曲线族上的平均值和受限x射线变换。此外，还研究了在积分几何和反地震问题中出现的退化傅里叶积分算子的种类，并给出了它们的组成演算。最小次代数变化的超平面截面Goncharov复形的实类似物和线性化源问题的正演算子将为此类算子提供重要的原型。本课题所分析的算子是一类特殊的傅立叶积分算子，它们在过去的三十年里一直被用来构造线性偏微分方程的近似解。这些方程控制着基本的物理过程，如膜的振动和电磁场的传播。最近，人们发现傅里叶积分算子出现在其他场景中，如断层扫描（医学成像）和地球物理勘探。在各种几何假设下更详细地了解这些操作符将提高我们对所建模的物理系统的定性和定量行为的理解。

Proposal: DMS-9877101Principal Investigator: Allan T. GreenleafAbstract: Estimates for Fourier integral operators associated with canonical relations whose projections exhibit stable classes of singularities, higher-order cusps in particular, will be investigated. For two-sided cusps, new decompositions of the operators near the singular varieties will be developed. These estimates and decompositions will have applications to boundedness properties of generalized Radon transforms, particularly averages over families of curves and restricted X-ray transforms, on Sobolev and Lebesgue spaces. In addition, classes of degenerate Fourier integral operators arising in integral geometry and inverse seismic problems will be studied and composition calculi obtained for them. Real analogues of Goncharov's complexes of hyperplane sections of algebraic varieties of minimal degree and the forward operator for linearized source problems will provide important prototypes of such operators.The operators analyzed in this project are special classes of Fourier integral operators, which have been used for the last thirty years to construct approximate solutions to linear partial differential equations. Such equations govern basic physical processes, such as vibration of membranes and propagation of electromagnetic fields. More recently, it has been found that Fourier integral operators arise in other settings, such as tomography (medical imaging) and geophysical exploration. More detailed understanding of these operators under a variety of geometric assumptions will improve our understanding of both the qualitative and quantitative behavior of the physical systems being modelled.