Layer potentials and essentially translation invariant pseudodifferential operators on manifolds with cylindrical ends

TL;DR

This paper develops new classes of pseudodifferential operators on manifolds with cylindrical ends, extending existing calculi, and proves their key properties, including spectral invariance, with applications to layer potentials and the Stokes operator.

Contribution

Introduction of two novel pseudodifferential calculi on manifolds with cylindrical ends, with spectral invariance and properties tailored for layer potential operators and elliptic problems.

Findings

The 'essentially translation invariant calculus' is spectrally invariant.

The calculi are stable under products and adjoints.

They possess desirable mapping, regularity, and Fredholm properties.

Abstract

Motivated by the study of layer potentials on manifolds with straight conical or cylindrical ends, we introduce and study two classes (or calculi) of pseudodifferential operators defined on manifolds with cylindrical ends: the class of pseudodifferential operators that are ``translation invariant at infinity'' and the class of ``essentially translation invariant operators.'' These are ``minimal'' classes of pseudodifferential operators containing the layer potential operators of interest. Both classes are close to the $b$-calculus considered by Melrose and Schulze and to the $c$-calculus considered by Melrose and Mazzeo-Melrose. Our calculi, however, are different and, while some of their properties follow from those of the $b$- or $c$-calculi, many of their properties do not. In particular, we prove that the ``essentially translation invariant calculus'' is spectrally invariant, a property not enjoyed by the ``translation invariant at infinity'' calculus or the $b$-calculus. For our calculi, we provide easy, intuitive proofs of the usual properties: stability for products and adjoints, mapping and boundedness properties for operators acting between Sobolev spaces, regularity properties, existence of a quantization map, topological properties of our algebras, and the Fredholm property. Since our applications will be to the Stokes operator, we systematically work in the setting of \ADN-elliptic operators. We also show that our calculi behave well with respect to restrictions to (suitable) submanifolds, which is crucial for our applications to layer potential operators.