Proper uniform pseudodifferential operators on unimodular Lie groups

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 onRn (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.