The Mellin-Edge Quantisation for Corner Operators

TL;DR

This paper develops a new Mellin-edge quantisation method for corner-degenerate operators on manifolds with second order singularities, extending previous edge quantisation techniques to more complex singularities.

Contribution

It introduces a novel quantisation framework for second order corner-degenerate symbols, addressing challenges from cone singularities and extending prior edge quantisation results.

Findings

Established Mellin-edge quantisation for second order singularities

Formulated operator-valued amplitude functions using Mellin symbols

Addressed new difficulties from cone singularities at higher order

Abstract

We establish a quantisation of corner-degenerate symbols, here called Mellin-edge quantisation, on a manifold $M$ with second order singularities. The typical ingredients come from the "most singular" stratum of $M$ which is a second order edge where the infinite transversal cone has a base $B$ that is itself a manifold with smooth edge. The resulting operator-valued amplitude functions on the second order edge are formulated purely in terms of Mellin symbols taking values in the edge algebra over $B.$ In this respect our result is formally analogous to a quantisation rule of a joint paper with J. Gil and J. Seiler for the simpler case of edge-degenerate symbols that corresponds to the singularity order 1. However, from the singularity order 2 on there appear new substantial difficulties for the first time, partly caused by the edge singularities of the cone over $B$ that tend to infinity.