9401855 Smith This award supports mathematical research focusing on applications of harmonic analysis to problems in the theory of hyperbolic differential equations. Work includes the development of estimates for the wave group with Dirichlet conditions on manifolds similar to that obtained with the additional condition that the boundaries be concave. Estimates fail on manifolds with convex boundaries although in these instances there exist multiply reflected geodesics producing gliding rays which travel along the boundary. It is therefore of interest to find worst case example where the estimates fail and try to obtain newer estimates for this situation. Work will also be done on the Hardy space for Fourier integral operators. Efforts will go into analyzing a new function space on which the algebra of Fourier integral operators of order zero act continuously. Certain operators associated with oscillatory problems are not bounded on the standard Hardy spaces will remain bounded on the new spaces. One of the first tasks is to obtain an alternate characterization of the space. Finally, work will be done constructing frames for square integrable functions which diagonalize the wave group. 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 wavelet theory added new dimensions to some of the more classical approaches to harmonic analysis.***
