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

Ville Turunen, Michael Ruzhansky

Research output: Contribution to journalArticleScientificpeer-review

49 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)2439-2496
Number of pages58
JournalINTERNATIONAL MATHEMATICS RESEARCH NOTICES
Volume2013
Issue number11
DOIs
Publication statusPublished - 2013
MoE publication typeA1 Journal article-refereed

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