Mathematical Sciences: Fourier Analysis

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Work supported by this award will focus on several areas of mathematical analysis related to singular integral operators, averages over curves and surfaces, maximal functions, Fourier integral operators, and Fourier multipliers on compact and noncompact manifolds. One of the central problems in mathematical analysis is that of recovering a function through various differentiation techniques applied to integrals of the function restricted to curves or surfaces. Although there is now a considerable body of literature on this subject, the curves have always been confined to those satisfying strong curvature conditions. Efforts to obtain more general information will require local maximal function estimates on compact sets. The thrust of the research will be to find these estimates first. This work, in turn, has close connections with singular integral operators of Calderon-Zygmund type. One of the obstructions is the dependence of past arguments on the Fourier transform. There is now some evidence that progress is possible which will bypass this impediment. Other work will concentrate on local smoothing properties for wave equations. The general idea behind local smoothing is one of establishing measurement of the size of a solution of a wave equation for time periods rather than for a fixed instant of time. The measurements are given in terms of the boundary values associated with the equation. Some local smoothing results have been obtained, especially in higher dimensions. The most difficult cases are spatial dimensions two and three. Ultimately, one must establish variants of the Kakeya- maximal operator and control of its p-th power norm. Work on estimates of degenerate Fourier integral operators, radial Fourier multipliers on noncompact Riemannian manifolds and estimates of the Szego kernel for pseudoconvex domains will continue. Mathematical research in harmonic analysis seeks to study the fine structure of mathematical functions and their underlying domains of definition. The tools developed by this work bring out hidden relationships not ordinarily detected by more classical methods.