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 Ψm(G). Properties of the new class and symbolic calculus are analyzed. Properties of symbols as well as L2-boundedness and Sobolev L2-boundedness of operators in this global quantization are established on general compact Lie groups. Operators on the three-dimensional sphere Graphic 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 Ψm(K\G) can be lifted to a pseudo-differential operator in Ψm(G), extending the known results on invariant partial differential operators.
Turunen, V., & Ruzhansky, M. (2013). Global quantization of pseudo-differential operators on compact Lie groups, SU(2), 3-sphere, and homogeneous spaces. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2013(11), 2439-2496. https://doi.org/10.1093/imrn/rns122