Research on this project will cover broad areas of modern mathematical analysis. Particular topics include harmonic analysis, singular integrals, quantum mechanics and several complex variables. Much of the work derives from geometric ideas. Included will be studies of singular Radon transforms along submanifolds in which the connection between boundedness of the transform and curvature properties of the submanifolds will be clarified. Significant progress has been made when the underlying structure is a nil-potent Lie group. Other work concerns smooth pseudoconvex domains in the space of several complex variables. Substantial advances have been made in understanding the nature of the two basic operators related to function theory studied on the domains: the Szego and Bergman projections. Holder estimates have been determined in the two- dimensional case. Further efforts will be made in eliminating the dimension requirement and finding sharp estimates for functions in Sobolev and Lebesgue spaces. Work is also continuing on a mathematical formulation providing a basis for the formation of atoms. The main issue is that of estimating the best constant in the stability of matter inequality. In addition, a program to identify all invariant polynomials attached to the Taylor expansion of a conformal metric at a point of a manifold will be continued. A complete list is believed to have been identified in all odd dimensions, but this has yet to have been verified. Somewhat related is work on the nodal sets of eigenfunctions of the Laplace operator on compact manifolds. It has been conjectured that these sets have codimension one Hausdorff measure on the order of the square root of the corresponding eigenvalue. This has now been established for manifolds with analytic metrics. Future work will concentrate on extending the result to metrics which are infinitely differentiable.
