In this paper, we use the representation theory of the group $$Spin(m)$$to develop aspects of the global symbolic calculus of pseudo-differential operators on $$Spin(3)$$and $$Spin(4)$$in the sense of Ruzhansky–Turunen–Wirth. A detailed study of $$Spin(3)$$and $$Spin(4)$$-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 $$Spin(4)$$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.
%0 Journal Article
%1 cerejeiras2023global
%A Cerejeiras, P.
%A Ferreira, M.
%A Kähler, U.
%A Wirth, J.
%D 2023
%J Journal of Fourier Analysis and Applications
%K from:jenswirth myown
%N 3
%P 32
%R 10.1007/s00041-023-10015-5
%T Global Operator Calculus on Spin Groups
%U https://doi.org/10.1007/s00041-023-10015-5
%V 29
%X In this paper, we use the representation theory of the group $$Spin(m)$$to develop aspects of the global symbolic calculus of pseudo-differential operators on $$Spin(3)$$and $$Spin(4)$$in the sense of Ruzhansky–Turunen–Wirth. A detailed study of $$Spin(3)$$and $$Spin(4)$$-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 $$Spin(4)$$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.
@article{cerejeiras2023global,
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.},
added-at = {2023-05-18T04:54:18.000+0200},
author = {Cerejeiras, P. and Ferreira, M. and Kähler, U. and Wirth, J.},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2771c7682a1db4f7a4d62a7a7f064311a/jenswirth},
doi = {10.1007/s00041-023-10015-5},
interhash = {76d31e1ddc26ece756158f8e427626ab},
intrahash = {771c7682a1db4f7a4d62a7a7f064311a},
issn = {15315851},
journal = {Journal of Fourier Analysis and Applications},
keywords = {from:jenswirth myown},
number = 3,
pages = 32,
refid = {Cerejeiras2023},
timestamp = {2023-05-18T06:26:47.000+0200},
title = {Global Operator Calculus on Spin Groups},
url = {https://doi.org/10.1007/s00041-023-10015-5},
volume = 29,
year = 2023
}
