Global Operator Calculus on Spin Groups

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