Operators with Corener-degenerate Symbols

TL;DR

This paper introduces a new approach to ellipticity and parametrices for operators on manifolds with higher singularities, using axiomatic frameworks and iterative operator theories.

Contribution

It develops a novel axiomatic method for ellipticity and parametrices, extending operator theories to higher singularities with new scales of spaces.

Findings

Established axiomatic framework for parameter-dependent operators

Analyzed symbols and operator classes near corner points

Proposed iterative process for higher singularity operator theories

Abstract

We establish elements of a new approch to ellipticity and parametrices within operator algebras on a manifold with higher singularities, only based on some general axiomatic requirements on parameter-dependent operators in suitable scales of spaces. The idea is to model an iterative process with new generations of parameter-dependent operator theories, together with new scales of spaces that satisfy analogous requirements as the original ones, now on a corresponding higher level. The "full" calculus is voluminous; so we content ourselves here with some typical aspects such as symbols in terms of order reducing families, classes of relevant examples, and operators near a corner point.