Mathematical Sciences: Variable Coefficient Fourier Analysis

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This award provides additional support for research by a Presidential Young Investigator working on problems of Fourier analysis related to the microlocal analysis of partial differential operators and the theory of Fourier integral operators. Work will be done in developing a microlocal version of the maximal theorem of Bourgain. The result concerns the maximal operator associated with the Radon transform (over spheres) which was shown to preserve the Lebesgue spaces for powers greater than two. The averaging may be viewed as a Fourier integral operator of negative order one-half. Efforts will be made to determine if the same result holds for any family of nondegererate operators of the same order, depending smoothly on a linear parameter. Some additional conditions have to be established since a total generalization is known to be impossible. A related type of operator studied by Carleson and Sjolin is known to preserve the fourth-power Lebesgue spaces. In this project, work will be done in expanding the operators to integral transforms involving exponential of the Laplace-Beltrami operator. If this is possible then the Carleson-Sjolin theorem for Bochner-Riesz summation in the plane can be extended to compact manifolds. Other work involves the question of embedded eigenvalues for Schrodinger operators. If the potential is bounded by the reciprocal of the variable, then Hormander showed that eigenfunctions which are small at infinity actually vanish everywhere. The next step in analyzing this phenomenon is to determine exact conditions on the potential to guarantee the same result. An immediate goal will be to find the correct function space (one of the local Lebesgue spaces) for embedding the potential.