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Operators with symbol hierarchies and iterated asymptotics. (English) Zbl 1051.58011

If one considers manifolds with sequences of degeneracies (stratified spaces) then, to establish a contentful elliptic theory, one has to consider a suitable hierarchy of symbols, associated with the system of lowerdimensional strata of the configuration. This philosophy has been highly elaborated by the author and is also the content of this paper. The author studies ellipticity on a manifold that has edges with conical singularities. Parametrices and iterated asymptotics of solutions to elliptic equations are determined by a three-component symbolic hierarchy, with interior, edge and conormal symbols. He constructs an operator algebra of \(2\times 2\)-block matrices, where the upper left corners contain the interior operators, together with so-called Green and Mellin operators (caused by analogues of Green’s function in boundary value problems as well as by asymptotic phenomena), while the other entries contain trace and potential conditions with respect to the edge and pseudo-differential operators on the edge itself that are of Fuchs type with respect to the conical points. The calculus is organized in an iterative way and can be viewed as a starting point for constructing similar operator algebras with asymptotics for higher polyhedral singularities.

MSC:

58J32 Boundary value problems on manifolds
58J40 Pseudodifferential and Fourier integral operators on manifolds
35J70 Degenerate elliptic equations
35S05 Pseudodifferential operators as generalizations of partial differential operators
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