Mathematical Sciences: Singular Integrals and Fourier Integral Operators

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This project seeks to analyze the mathematical theories of singular integral, Fourier integral and pseudodifferential operators. The theories developed over the past thirty years yielding a wide range of applications based on their simplicity and unity. Nevertheless, operators falling outside their scope are appearing with increased frequency in studies of Fourier-Airy operators of diffraction theory, maximal operators and Hilbert transforms along curves of differentiation theory, Hilbert integral operators and singular Radon transforms of subelliptic boundary value problems, oscillatory singular integrals on nilpotent groups, the X-ray transforms and Guillemin operators of integral geometry and deformation theory on manifolds of geodesics. Two unifying themes have emerged; they appear themselves to be related. The first consists of Fourier integral operators whose Lagrangian is not a local graph but projects on tangent spaces with folds or cusps. The second is where the densities on the Lagrangian have singularities. The main goal of this work is to arrive at a more complete theory of degenerate Fourier integral and singular integral operators. Strong evidence for the existence of such a framework is provided by recent progress in several directions: work will be done in exploring them further. First, efforts will be made to understand the stratification of the singular varieties of projections of degenerate operators on tangent spaces and how they influence Sobolev bounds. This will only treat operators with constant coefficients. Efforts at a formulation of the theory on manifolds, a new phenomenon - not present in the classical case - occurs. Work must be done in obtaining lower bounds for the oscillation of the phase function. Other goals involve finding extensions to operators defined in spaces of higher dimension and the measure of the level of stratification of the local geometry of the Lagrangian to yield sharp bounds for singular Radon transforms.