Several basic mathematical questions in Fourier analysis on Euclidean space will be taken up in the project. The objectives are to analyze the decay of Fourier transforms of measures concentrated on surfaces and power-norm estimates for singular integral and maximal operators associated with curves. This work derives from fundamental questions concerning partial differential operators defined on various function spaces. Fourier transforms of functions restricted to hypersurfaces have controlled asymptotic growth. Estimates for this growth (or decay) have been the subject of intense research and are known to run from absence of decay (for hyperplanes) to reciprocal power decay for surfaces of finite type. This type of decay is not restricted to such surfaces although the order of contact of a surface with its tangent plane plays some, as yet unknown, major role in guaranteeing decay at infinity (of the transform). Work will be done in finding the correct conditions between finite type and "flat" surfaces. Recent unpublished examples have confirmed that convexity is not the right criterion. A related line of investigation concerns the Hilbert transform and maximal function along a convex plane curve. Of concern are the power-norms of these functions and their relationship to the same norms of the antecedent function. The theory is relatively well developed for convex curves given by smooth even functions but breaks down when these conditions are weakened. Efforts to find the exact conditions for boundedness of the two transforms will continue.
