Complex powers and non-compact manifolds

TL;DR

This paper develops an axiomatic framework for complex powers of elliptic pseudodifferential operators using extended Weyl algebras, enabling precise estimates on non-compact manifolds with Lie structures at infinity.

Contribution

It introduces extended Weyl algebras as a unifying framework for pseudodifferential operators, generalizing key properties and enabling analysis on non-compact manifolds.

Findings

Extended Weyl algebras encompass many pseudodifferential operator algebras.

Results include asymptotic completeness and Sobolev space constructions.

Framework allows precise estimates of complex powers on non-compact manifolds.

Abstract

We study the complex powers $A^{z}$ of an elliptic, strictly positive pseudodifferential operator $A$ using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, ``extended Weyl algebras,'' whose definition was inspired by Guillemin's paper on the subject. An extended Weyl algebra can be thought of as an algebra of ``abstract pseudodifferential operators.'' Many algebras of pseudodifferential operators are extended Weyl algebras. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between apropriate Sobolev spaces, >...) generalize to extended Weyl algebras. Most important, our results may be used to obtain precise estimates at infinity for $A^{z}$, when $A > 0 $ is elliptic and defined on a non-compact manifold, provided that a suitable ideal of regularizing operators is specified (a submultiplicative $\Psi^*$--algebra). We shall use these results in a forthcoming paper to study pseudodifferential operators and Sobolev spaces on manifolds with a Lie structure at infinity (a certain class of non-compact manifolds).