Global quantization of pseudo-differential operators on compact Lie groups, SU(2) and 3-sphere

TL;DR

This paper develops a global quantization framework for pseudo-differential operators on compact Lie groups, especially on SU(2) and the 3-sphere, using representation theory instead of local coordinates.

Contribution

It introduces a new class of globally defined symbols for pseudo-differential operators on compact Lie groups, extending Hormander's classes and analyzing their properties and boundedness.

Findings

Established properties of the new symbol class.

Proved $L^2$-boundedness and Sobolev boundedness of operators.

Analyzed symbolic calculus on SU(2) and the 3-sphere.

Abstract

Global quantization of pseudo-differential operators on compact Lie groups is introduced relying on the representation theory of the group rather than on expressions in local coordinates. Operators on the 3-dimensional sphere and on group SU(2) are analysed in detail. A new class of globally defined symbols is introduced giving rise to the usual Hormander's classes of operators $\Psi^m(G)$, $\Psi^m(S^3)$ and $\Psi^m(SU(2))$. Properties of the new class and symbolic calculus are analysed. Properties of symbols as well as $L^2$-boundedness and Sobolev $L^2$--boundedness of operators in this global quantization are established on general compact Lie groups.