Mellin and Green pseudodifferential operators associated with non-compact edges
The author considers an algebra of global wedge pseudo-differential operators on the infinite (open stretched) wedge \(\mathbb{R}^q\times X^\wedge\) where \(X^\wedge=\mathbb{R}_+\times X\) is the open (stretched) cone with base \(X\), closed compact smooth manifold. With respect to the theory of \textit{B.-W. Schulze} [Pseudo-differential operators on manifolds with singularities, North-Holland (1991; Zbl 0747.58003), Pseudo-differential boundary value problems, conical singularities and asymptotics, Akademie Verlag, Berlin (1994; Zbl 0810.35175)] concerning manifolds with compact edges, here a suitable structure of the symbol is required at the non-compact edge; the author refers in particular to the pseudo-differential calculus of \textit{C. Parenti} [Ann. Math. Pura appl., IV. Ser. 93, 359-389 (1972; Zbl 0291.35070)] and \textit{H. O. Cordes} [The technique of pseudo-differential operators, Cambridge University Press (1995; Zbl 0828.35145)]. In this frame the author is able to establish a notion of ellipticity that incorporates the corresponding concept for compact edges, and is equivalent to the existence of a parametrix within the calculus. The exposition is self-contained and presents some interesting improvements of the vector-valued symbols technique.
global wedge pseudo-differential operators
scientific article; zbMATH DE number 1187566