Cone pseudodifferential operators in the edge symbolic calculus

Abstract

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. Our results fit into the frame of pseudodifferential calculi on manifolds with singularities, particularly with piecewise smooth geometry. They belong to the idea to reflect the stratification of such a space 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. It is well-known that, for instance, boundary value problems for pseudodifferential operators can be represented as operators along the boundary with operator-valued symbols, acting along R+, the inner normal. In this sense, not only Boutet de MonveΓs algebra [1] (cf. also Schrohe and Schulze [7]) of boundary value problems with the transmission property is included in the context but also Vishik, Eskin's theory [3], [12], turned into a corresponding operator algebra, cf. Schulze [10]. The iteration of calculi leads to very complex analytic phenomena, and it is still a serious problem to formulate manageable operator algebras for higher singularities such as of corner type or for boundary value problems with such singular boundaries. The main objective of our paper is to develop an efficient new approach to the algebra of cone operator-valued edge symbols as originally introduced in [9]. 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 (cf. [2], [10]). In the new representation we 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 dismissed. This relies on a form of the Mellin quantization for