Elliptic parametrices in the 0-calculus of Mazzeo and Melrose

TL;DR

This paper details the construction of parametrices for fully elliptic degenerate pseudodifferential operators on manifolds with boundary, utilizing the 0-calculus framework and advanced geometric resolutions.

Contribution

It explicitly constructs parametrices with polyhomogeneous conormal kernels using the 0-double space and extends the original 0-calculus with Lauter's extended double space.

Findings

Parametrices have polyhomogeneous conormal Schwartz kernels.

The extended 0-double space facilitates the parametrix construction.

The approach generalizes Mazzeo-Melrose's 0-calculus to broader settings.

Abstract

The purpose of this note is to spell out the details of the construction of parametrices for fully elliptic uniformly degenerate pseudodifferential operators on manifolds $X$ with boundary. Following the original work by Mazzeo-Melrose on the 0-calculus, the parametrices are shown to have (polyhomogeneous) conormal Schwartz kernels on the 0-double space, a resolution of $X^2$. The extended 0-double space introduced by Lauter plays a useful role in the construction.