An Approach to a Version of the S(M, g)-pseudo-differential Calculus on Manifolds

Abstract

For appropriate triples (M,g,M), where M is an (in general non-compact) manifold, g is a metric on T* M, and M is a weight function on T* M, we develop a pseudo-differential calculus on.A4 which is based on the S(M,g))-calculus of L. Hörmander [27] in local models. In order to do so, we generalize the concept of E. Schrohe [41] of so-called SG-compatible manifolds. In the final section we give an outlook onto topological properties of the algebras of pseudo-differential operators. We state the existence of “order reducing operators” and that the algebra of operators of order zero is a submultiplicative Ψ*-algebra in the sense of B. Gramsch [18] in \( \mathcal{L}\left( {{L^2}\left( M \right)} \right) \).Key words and phrasespseudo-differential operators on manifoldsWeyl-Hörmander calculusSG-calculusspectral invariancesubmultiplicativity2000 Mathematics Subject ClassificationAMS 1991 Subject ClassificationPrimary 35S0558G15 Secondary 47A6047A1047D3047F05