Operator Algebras with Symbol Hierarchies on Manifolds with Singularities

Abstract

Problems for elliptic partial differential equations on manifoldsMwith singularitiesM’(here with piece—wise smooth geometry) are studied in terms of pseudo-differential algebras with hierarchies of symbols that consist of scalar and operator—valued components. Classical boundary value problems (with or without the transmission property) belong to the examples. They are a model for operator algebras on manifoldsMwith higher “polyhedral” singularities. The operators are block matrices that have upper left corners containing the pseudo—differential operators on the regular part M\M’ (plus certain Mellin and Green summands) and are degenerate (in stretched coordinates) in a typical way nearM’.By definitionM’is again a manifold with singularities. The same is true ofM“, and so on. The block matrices consist of trace, potential and Mellin and Green operators, acting between weighted Sobolev spaces onM()andM( 0 ),with 0 <j k <ord M; hereMO)denotesM, MO) denotesM’,etc. We generate these algebras, including their symbol hierarchies, by iterating so—called “edgifications” and “conifications” of algebras that have already been constructed, and we study ellipticity, parametrices, and Fredholm property within these algebras.