The Mellin Pseudo-Differential Calculus on Manifolds with Corners

Differential and pseudo-differential operators on a manifold with (regular) geometric singularities can be studied within a calculus, inspired by the concept of classical pseudo-differential operators on a C ∞ manifold. In the singular case the operators form an algebra with a principal symbolic hierarchy σ = (σj)0≤j≤k, with k being the order of the singularity and σk operator-valued for k ≥ 1. The symbols de-termine ellipticity and the nature of parametrices. It is typical in this theory that, similarly as in boundary value problems (which are special edge problems, where the edge is just the boundary), there are trace, potential and Green operators, associated with the various strata of the configuration. The operators, obtained from the symbols by various quantisations, act in weighted distribution spaces with multiple weights. We outline some essential elements of this calculus, give ex-amples and also comment on new challenges and interesting problem