Mellin Operators in the Edge Calculus

A manifold M with smooth edge Y is locally near Y modelled on \(X^\vartriangle \times {\varOmega }\) for a cone \(X^\vartriangle :=(\overline{{\mathbb {R}}}_+\times X)/(\{0\}\times X)\) where X is a smooth manifold and \({\varOmega } \subseteq {\mathbb {R}}^q\) an open set corresponding to a chart on Y. Compared with pseudo-differential algebras, based on other quantizations of edge-degenerate symbols, we extend the approach with Mellin representations on the r half-axis up to \(r=\infty ,\) the conical exit of \(X^{\wedge }={\mathbb {R}}_+\times X\ni (r,x)\) at infinity. The alternative description of the edge calculus is useful for pseudo-differential structures on manifolds with higher singularities.