Mathematical Sciences: Oscillatory Integrals in Harmonic Analysis and Their Applications

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9401277 Pan This award supports mathematical research on problems arising in the theory of harmonic analysis. The work revolves around ideas relating to oscillatory integrals and their applications. Oscillatory integrals are an essential part of harmonic analysis. There are two kinds. The first involves a single function acting as a generalized Fourier transform with nonlinear exponent. The second is given by an integral operator with a kernel carrying oscillatory factors. This project involves studies of both kinds. An important class of oscillatory integrals of the second kind is the class oscillatory singular integrals. As a combination of oscillatory integral and singular integral, these operators are intimately connected to problems concerning the convergence of Fourier series, singular harmonic analysis on the Heisenberg group and other nilpotent Lie groups and singular Radon transforms. The project involves a systematic study of oscillatory integrals to examine their boundedness properties when applied to Lebesgue spaces, weak (1,1)-boundedness, boundedness on Hardy spaces, optimal decay in the quadratic norm and applications to convolution with measures supported on submanifolds. Harmonic analysis combines those elements of mathematics best exemplifying the ideas of synthesis. One seeks to decompose complex problems into fundamental components. These components are then analyzed for their basic characteristics. Finally, the solution is reconstructed through a recombination of the components. The Fourier series and Fourier transform are examples of tools used in this context; one discrete , the other representing a continuous decomposition. More recently the theory of oscillatory integrals added new dimensions to some of the more classical approaches to harmonic analysis. ***