Global Quantization of Pseudo-Differential Operators on Compact Lie Groups, SU(2), 3-sphere, and Homogeneous Spaces

Global quantization of pseudo-differential operators on general compact Lie groups G is introduced relying on the representation theory of the group rather than on expressions in local coordinates. A new class of globally defined symbols is introduced and related to the usual Hörmander’s classes of operators Ψ
^m
(G). Properties of the new class and symbolic calculus are analyzed. 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. Operators on the three-dimensional sphere
$\mathbb S^3$
and on group SU(2) are analyzed in detail. An application is given to pseudo-differential operators on homogeneous spaces K\G. In particular, using the obtained global characterization of pseudo-differential operators on Lie groups, it is shown that every pseudo-differential operator in Ψ
^m
(K\G) can be lifted to a pseudo-differential operator in Ψ
^m
(G), extending the known results on invariant partial differential operators.