This Presidential Young Investigator Award will support mathematical research in the area of harmonic analysis related to problems of oscillatory integrals and differentiation theory. Several important directions will be followed during the period of support. Many of these have their roots in questions concerning partial differential operators and efforts to measure bounds on their solutions. These include bounds on the norm of spectral projections involving spherical harmonic expansions of functions on a sphere. Other work involves problems of Fourier analysis related to curvature of manifolds. This includes development of uniform Sobolev inequalities and corresponding restriction theorems. Results along these lines will provide powerful global conclusions from local inequalities. Work will continue on the question of sharp Riesz means on arbitrary compact manifolds, especially regarding the weak type end point results which are invariably the most difficult to achieve. Other work will focus on unique continuation of second order partial differential operators, especially Schrodinger operators, unique continuation of hyperbolic and parabolic differential operators and related eigenfunction expansions. A new area of work will concern maximal estimates for Randon transforms or averages of functions over lower dimensional surfaces. Interestingly the most difficult questions in this context occur in lower dimensions where there has been considerable activity lately.
