Global Operator Calculus on Spin Groups

Abstract

In this paper, we use the representation theory of the group textrm{Spin}(m) to develop aspects of the global symbolic calculus of pseudo-differential operators on textrm{Spin}(3) and textrm{Spin}(4) in the sense of Ruzhansky–Turunen–Wirth. A detailed study of textrm{Spin}(3) and textrm{Spin}(4)-representations is made including recurrence relations and natural differential operators acting on matrix coefficients. We establish the calculus of left-invariant differential operators and of difference operators on the group textrm{Spin}(4) and apply this to give criteria for the subellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of first and second order globally hypoelliptic differential operators are given, including some that are locally neither invertible nor hypoelliptic. The paper presents a particular case study for higher dimensional spin groups.