Elliptic operators on refined Sobolev scales on vector bundles

TL;DR

This paper develops a new refined Sobolev scale on vector bundles over smooth manifolds, analyzing elliptic pseudodifferential operators within this framework, and establishing boundedness, Fredholm properties, and regularity results for solutions.

Contribution

It introduces a novel refined Sobolev scale based on Hörmander spaces parametrized by a real number and a slowly varying function, and studies elliptic operators on this scale.

Findings

Elliptic operators are bounded and Fredholm on the refined scale.

Solutions satisfy specific a priori estimates in these spaces.

New conditions for solutions to have continuous derivatives are established.

Abstract

We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product H\"ormander spaces parametrized with a real number and a function varying slowly at infinity in the sense of Karamata. We prove that these spaces are obtained by the interpolation with a function parameter between inner product Sobolev spaces. An arbitrary classical elliptic pseudodifferential operator acting between vector bundles of the same rank is investigated on this scale. We prove that this operator is bounded and Fredholm on pairs of appropriate H\"ormander spaces. We also prove that the solutions to the corresponding elliptic equation satisfy a certain a priori estimate on these spaces. The local regularity of these solutions is investigated on the refined Sobolev scale. We find new sufficient conditions for the solutions to have continuous derivatives of a given order.