Mathematical Sciences: Singular Integrals and Fourier Integrals

在这个项目中，首席研究员将研究与奇异积分和傅立叶积分相关的几个问题。特别是，他将建立一类适用于积分几何和等谱变形实例的正则关系，并证明与之相关的傅里叶积分算子的复合微积分和L-2估计。主要研究者还将研究一阶伪微分算子组的局部可解性。最后，他将得到带奇异符号的退化拟微分算子的更精确的估计，并将这些结果应用于退化的Radon变换。理论和应用分析中的许多问题都与运算或变换的求逆有关。例如，一个问题的期望解可能被转换成一个涉及积分或积分算子的方程，然后为了获得解，必须逆变换，也就是解积分方程式。在这个项目中，首席研究员将分析两类积分算子，以便对它们所代表的解提供准确的估计。这项工作的一个应用是估计X射线层析成像问题的解。

In this project the principal investigator will study several problems associated with singular integrals and Fourier integrals. In particular, he will formulate classes of canonical relations that apply to examples from integral geometry and isospectral deformations, and he will prove composition calculi and sharp L-2 estimates for Fourier integral operators associated with them. The principal investigator will also look into the local solvability of systems of first-order pseudodifferential operators. Finally he will obtain sharper estimates for degenerate pseudodifferential operators with singular symbols, and he will apply these results to degenerate Radon transforms. A number of problems in theoretical and applied analysis concern the inversion of an operation or a transformation. For example, the desired solution of a problem may be transformed into an equation that involves integrals or integral operators, and then in order to obtain the solution, one must invert the transformation, that is, solve the integral equation. In this project the principal investigator will analyze two classes of integral operators, in order to provide sharp estimates for the solutions they represent. One application of this work is to the estimation of solutions of X-ray tomography problems.