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.
