Donal Connelly (Imperial) - "Pseudo-differential operators on homogeneous spaces"

Abstract

In the theory of PDE, pseudo-differtial operators (ΨDO) arise as a natural way to study the properties of solutions to certain PDE. A Euclidean ΨDO of order m is a generalization of a PDO of order m.

A homogeneous space is the quotient space G/K, where G is a compact Lie group and K a closed subgroup. One can define the Hörmander classes of ΨDO of order m on G/K as those continuous linear operators on C^∞(G/K) whose localizations are Euclidean ΨDO of order m.

In this talk, we will use Peter-Weyl theory (the theory of Fourier analysis on compact groups) to associate "global symbols'' to operators on C^∞(G/K). We will then use these symbols to characterize the Hörmander classes of ΨDO on G/K.