Mathematical Sciences: Singular Integrals and Fourier Integrals

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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.