The present paper studies an algebra of parameter-dependent pseudodifferential operators on a manifold with conical singularities, where the parameters are involved as covariables in a specific degenerate way. Such operator families serve as an adequate symbol class for pseudodifferential operators on manifolds with edges. Also, the study of resolvents of differential operators of Fuchs type leads to families of a similar form.
The results belong to the idea to reflect the stratification of a manifold with singularities by a hierarchy of operator algebras with symbolic structure, and to organize an iterative procedure which starts from the calculus on a given space, say a cone, and constructs a next “higher” calculus on a space with higher-order singularities, say a wedge.
The main objective of this paper is to develop an efficient new approach to the algebra of cone operator-valued edge symbols as originally introduced in [B. W. Schulze, Teubner-Texte Math. 112, 259-287 (1989; Zbl 0693.58026)]. One of the difficulties is that the edge covariables are involved in a degenerate form, i.e. multiplied by the axial variable of the cone. In the new representation, the authors can, in particular, avoid a number of extremely voluminous calculations in the precise analysis of operator-valued edge symbols by a new quantization of edge-degenerate interior symbols in which a part of the inconvenient combinations of edge covariable and axial variable is discussed. This relies on a form of Mellin quantization for edge-degenerate pseudodifferential symbols.