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 Hormander's classes of operators Psi(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 S-3 and on group SU(2) are analyzed in detail. An application is given to pseudo-differential operators on homogeneous spaces K backslash G. In particular, using the obtained global characterization of pseudo-differential operators on Lie groups, it is shown that every pseudo-differential operator in Psi(m)(K backslash G) can be lifted to a pseudo-differential operator in Psi(m)(G), extending the known results on invariant partial differential operators.