Fourier Integrals and Generalized Radon Transforms

提案：DMS-9877101主要研究者：Allan T. Greenleaf摘要：估计傅立叶积分算子与典型的关系，其投影表现出稳定类的奇点，特别是高阶尖点，将被调查。对于双边尖点，新的分解附近的奇异品种的运营商将开发。这些估计和分解将应用于广义Radon变换的有界性，特别是平均家庭的曲线和限制X射线变换，Sobolev和Lebesgue空间。此外，在积分几何和地震反问题中出现的退化傅里叶积分算子类将被研究，并获得它们的合成演算。Goncharov复形的最小代数簇的超平面部分的真实的类似物和线性化源问题的正向算子将提供这种算子的重要原型。在这个项目中分析的算子是特殊类的傅里叶积分算子，在过去的三十年中，它们被用来构造线性偏微分方程的近似解。这样的方程支配着基本的物理过程，例如膜的振动和电磁场的传播。最近，人们发现傅立叶积分算子出现在其他设置，如断层摄影（医学成像）和地球物理勘探。在各种几何假设下对这些算子的更详细的理解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.