The local smoothing conjecture for the Euclidean wave equation is a major open problem in harmonic analysis. In this talk I will discuss some recent work connecting local smoothing estimates and invariant spaces for Fourier integral operators. The resulting estimates improve or complement those in the local smoothing conjecture, and they are essentially sharp. I will also indicate how local smoothing estimates can be used to obtain new wellposedness results for nonlinear wave equations with slowly decaying initial data.
This talk is based on joint work with Robert Schippa (Karlsruhe Institute of Technology), and Naijia Liu, Liang Song and Lixin Yan (Sun Yat-Sen University).