A Shubin pseudodifferential calculus on asymptotically conic manifolds

TL;DR

This paper develops a global pseudodifferential calculus tailored for asymptotically conic manifolds, extending classical Euclidean calculus to noncompact settings and establishing Fredholm properties for elliptic operators.

Contribution

It introduces a new pseudodifferential calculus on asymptotically conic manifolds, generalizing Shubin's calculus and proving elliptic operators are Fredholm with parametrices.

Findings

Fully elliptic operators are Fredholm in Sobolev spaces.

Elliptic operators admit parametrices within the calculus.

The calculus extends classical Euclidean pseudodifferential methods.

Abstract

We present a global pseudodifferential calculus on asymptotically conic manifolds that generalizes (anisotropic versions of) Shubin's classical global pseudodifferential calculus on Euclidean space to this class of noncompact manifolds. Fully elliptic operators are shown to be Fredholm in an associated scale of Sobolev spaces, and to have parametrices in the calculus.