Abstract
An algebra of proper pseudodifferential operators on an arbitrary unimodular Lie group is constructed. This algebra is a generalization of a well-known algebra of operators with uniform estimates of the symbols onR n (such operators have been investigated in detail by Kumano-go); in the general case the estimates have to be left-invariant. An L2-boundedness theorem is proved and uniform Sobolev spaces are introduced and investigated. The essential self-adjointness of uniformly elliptic operators is proved. A criterion for the coincidence of the “left” and “right” Sobolev spaces and of the corresponding algebras of operators is given: it is necessary and sufficient that the considered Lie group be a central extension of a compact group.
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Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 74–97, 1986.
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Meladze, G.A., Shubin, M.A. Proper uniform pseudodifferential operators on unimodular Lie groups. J Math Sci 45, 1421–1439 (1989). https://doi.org/10.1007/BF01097159
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DOI: https://doi.org/10.1007/BF01097159