Oscillatory Integrals: Generalized Radon Transforms and Inverse Problems

将研究三个问题，它们与退化（或奇异）相函数的振荡积分的共同线索有关。首先，将考虑与对称和非对称的双边Morin奇点相关的傅里叶积分算子的估计。这将通过使用反映相函数的基本几何的算子的新分解来尝试。其次，对于大于2维的低规则电导率和电势，在Calderon问题中考虑全局唯一性，使用Radon变换分解电势。最后，将对二维低规则和/或正常电导率的问题进行详细的微局部分析。本项目研究的算子将一个空间上的函数转换为另一个空间上的函数（可能相同，也可能不同），其中振荡（由高频波引起的抵消）起着重要作用。在过去的三十年中，这种一般类型的算子在理解广泛的偏微分方程方面取得了进展，例如，声波和电磁波的波传播。事实证明，CAT扫描仪和其他非侵入性医学成像技术的数学分析也属于同一框架。当前的项目应该更好地理解这些技术的数学基础。从长远来看，这可能有助于这些成像方法的改进和改进。

Proposal Number: DMS-0138167PI: Allan GreenleafABSTRACT Three problems will be investigated, related by thecommon thread of oscillatory integrals with degenerate (or singular) phase functions. First, estimates for Fourierintegral operators associated with two-sided Morin singularities, both symmetric and asymmetric, will beconsidered. This will be attempted by using new decompositions of the operators that reflect the underlyinggeometry of the phase functions. Secondly, global uniquenesswill be considered in the Calderon problem for low-regularityconductivities and potentials in dimensions greaterthan two, using Radon transform decompositions of the potentials. Finally, a detailed microlocal analysis will be made of the problem for low-regularity and/or conormal conductivities in two dimensions. The operators to be studied in this project transformfunctions on one space to functions on another space (possibly the same, possibly different) in a way in whichoscillation (cancellation caused by high-frequency waves)plays an important role. This general type of operator hasbeen central to progress in the last thirty years in theunderstanding of a wide range of partial differentialequations governing, e.g., wave propagation of sound andelectromagnetic waves. It turns out that the mathematicalanalysis of CAT scanners and other noninvasive medical imaging technologies falls under the same framework.The current project should give a better understanding of themathematical foundations of these techniques. In the longerterm, this might contribute to refinement and improvementof these imaging methods.