Sobolev Spaces on Vector Bundles

Abstract

The aim of this chapter is to establish the basics corresponding to differential operators that act on sections of vector bundles, and to introduce some induced abstract Sobolev-type spaces. Clearly, this requires some understanding of distributional or weak derivatives. In principle, such weak derivatives can be defined with the help of an intrinsic theory of distributions, in which the space of test sections of a smooth vector bundle \(E\rightarrow X\) is given by the locally convex space of smooth compactly supported sections of \(E^*\otimes |X| \rightarrow X\).