Mathematical Sciences: Singular Integrals and Fourier Integrals

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The mathematical theme of this research is a combination of harmonic analysis and the modern theory of differential operators applied to questions about integral transforms of certain type. A physical counterpart to this analysis occurs in x-ray reconstruction or tomography where one seeks to visualize an unknown object by information derived from lower dimensional pictures taken from a limited number of angles. In the mathematical setting, it is assumed that a class of surface averages (Radon transformations) are known, where the surface here may be an abstract manifold, and form what is called a Fourier integral operator. Efforts will be made to obtain estimates for such averages or transforms in what is called the singular case. Examples in this particular context occur naturally when one considers Hilbert transforms along curves or operators associated with the d-bar Neumann problem on weakly pseudoconvex domains in several complex variables. When the transforms are not degenerate, mean-square estimates can be extended to all power norms. This work will concentrate on the degenerate case where estimates are harder to obtain and, in fact, the norm estimates for Radon transforms may fail, except in the mean-square case. Work will also be done in an attempt to formulate classes of Fourier integral operators sufficient to deal with the problem of admissibility and invertibility for the restricted k-plane transform in Euclidean space. The construction of the relevant canonical relations should be related to the Grassmannian geometry (of the k-planes).