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

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 ${\mathrm{\Psi }}^m(G)$, ${\mathrm{\Psi }}^m(S^3)$ and ${\mathrm{\Psi }}^m(SU(2))$. 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.