Mathematical Sciences: Singular Integrals and Fourier Integrals

某些与局部正则图无关的傅里叶积分算子在调和分析、积分几何和偏微分方程中自然出现。这个项目将制定正则关系的类别，并找到与这些类别相关的傅立叶积分算子的组合演算和尖锐估计。最简单的例子是，当人们试图确定算子的组合是否仍然是傅里叶积分算子，而没有Hormander的横交假设或Duistermaat和Guillemin提出的干净的交演算时。特别地，我们将研究退化算子的情况。重点将放在那些在相位中具有三次简并的振荡积分模型上，目的是获得估计和组成性质。应用包括估计和非局部反演公式的限制两平面（和，可能，k平面）变换。结合与奇异拉格朗日量相关的广义傅里叶积分分布，分析了实n维空间上奇异共形度量的后向散射。本研究的基本目标是通过反转加分解的基本技术来理解复杂的微分和积分算子。这项工作结合了分析和几何中强大的抽象方法来开发底层结构，其中建立了操作员扫描的属性。

Certain Fourier integral operators which are not associated with local canonical graphs arise naturally in harmonic analysis, integral geometry and partial differential equations. This project will formulate classes of canonical relations and find composition calculi and sharp estimates for Fourier integral operators associated with these classes. The simplest example arises when one seeks to determine whether the composition of operators is again a Fourier integral operator without the transverse intersection hypothesis of Hormander or the clean intersection calculus developed by Duistermaat and Guillemin. In particular work will be done on the case of degenerate operators. Emphasis will be placed on those modelled on oscillatory integrals with cubic degeneracies in the phase with the goal of obtaining estimates and composition properties. Applications include estimates and nonlocal inversion formulae for restricted two-plane (and, possibly, k-plane) transforms. In connection with classes of generalized Fourier integral distributions associated with singular Lagrangians, the backscattering by a singular conformal metric on real n-dimensional space will be analyzed. The underlying objectives of this research are to understand complex differential and integral operators through fundamental techniques of inversion plus decomposition. The work combines powerful abstract methods in analysis and geometry to develop underlying structures within which properties of the operator scan be established.