Spectral Calculus, Besov Spaces and Dispersive Equations

In this talk, Dr. Zheng will consider Hoermander type spectral multiplier problem for Schroedinger operators with a critical potential. It is shown that the multiplier operator is bounded on L^p, Besov and Triebel-Lizorkin spaces under the same sharp condition. The approach is an improvement upon Hebisch's heat kernel approach for nonnegative potentials in 1990. We then derive Strichartz estimates that measure spacetime regularity for the corresponding wave equation. Our work is partially motivated by the standing wave problem for the quintic wave equation in 3+1 dimensions.