Abstract
AbstractIn this paper, we use the representation theory of the group $$\textrm{Spin}(m)$$
Spin
(
m
)
to develop aspects of the global symbolic calculus of pseudo-differential operators on $$\textrm{Spin}(3)$$
Spin
(
3
)
and $$\textrm{Spin}(4)$$
Spin
(
4
)
in the sense of Ruzhansky–Turunen–Wirth. A detailed study of $$\textrm{Spin}(3)$$
Spin
(
3
)
and $$\textrm{Spin}(4)$$
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 $$\textrm{Spin}(4)$$
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.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,General Mathematics,Analysis
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