Analytic index formulas for elliptic corner operators
This paper develops an approach to the index theory of elliptic pseudodifferential operators on manifolds with corners. A corner is a cone over a manifold with conical points. The pseudodifferential operators live in the corner algebra introduced by \textit{B.-W. Schulze} [The Mellin pseudodifferential calculus on manifolds with corners, in Schulze, B.-W. (ed.) et al., Symposium ``Analysis on manifolds with singularities,'' Teubner-Texte Math. 131, 208-289 (1992; Zbl 0810.58041)]. (These operators are not the totally characteristic operators introduced by R. Melrose.) A corner operator is called elliptic if its principal interior symbol, principal edge symbol, and principal corner symbol are invertible. An elliptic corner operator defines a Fredholm map between appropriate weighted Sobolev spaces if the corner conormal symbol satisfies a holomorphy condition. In the absence of the holomorphy condition, the authors show how to modify the operator to obtain an operator satisfying the holomorphy condition. The authors establish an index formula that applies to the elliptic operator or its modification, as appropriate. Operator-valued symbols play a role in the analysis of pseudodifferential operators on manifolds with corners. The authors view the calculus of parameter-dependent operator-valued symbols from a perspective in which domain and range norms vary with the parameter. They suggest that this perspective will simplify the treatment of more complicated singularities. The authors (alone, in pairs, or as a threesome) are responsible for the development of much of the area this paper sits in.
scientific article; zbMATH DE number 1794819
scientific article