Quasi-fibered boundary pseudodifferential operators

TL;DR

This paper develops a pseudodifferential calculus for quasi-fibered boundary metrics, establishing Fredholm properties of associated operators, constructing parametrices for the Hodge-deRham operator, and analyzing the decay of harmonic forms.

Contribution

It introduces a new calculus for QFB metrics, defines fully elliptic operators, and constructs parametrices to study their Fredholm properties and harmonic form decay.

Findings

QFB operators are Fredholm on Sobolev spaces.

Constructed parametrices for the Hodge-deRham operator.

Proved faster decay of $L^2$ harmonic forms at infinity.

Abstract

We develop a pseudodifferential calculus for differential operators associated to quasi-fibered boundary metrics (QFB metrics), a class of metrics including the quasi-asymptotically conical metrics (QAC metrics) of Degeratu-Mazzeo and the quasi-asymptotically locally Euclidean metrics (QALE metrics) of Joyce. Introducing various principal symbols, we introduce the notion of fully elliptic QFB operators and show that those are Fredholm when acting on QFB Sobolev spaces. For QAC metrics, we also develop a pseudodifferential calculus for the conformally related class of Qb metrics. We use these calculi to construct a parametrix for the Hodge-deRham operator of certain QFB metrics, allowing us to show that it is Fredholm on suitable Sobolev spaces and that the space of $L^2$ harmonic forms is finite dimensional. Our parametrix is obtained by inverting certain model operators at infinity, inversions that we achieve in part through a fine understanding of the low energy limit of the resolvent of the Hodge-deRham operator. Our parametrix also implies that $L^2$ harmonic forms decay faster at infinity than an arbitrary $L^2$ form, the extra decay being quantified in terms of a small negative power of the distance function. This decay of $L^2$ harmonic forms is used in a companion paper to study the $L^2$ cohomology of some $QFB$ metrics.