Analytic and elliptic estimates on non-compact manifolds via weighted Sobolev spaces

TL;DR

This paper provides a comprehensive, accessible overview of weighted Sobolev spaces and elliptic operators on non-compact manifolds, filling gaps in the literature and aiding applications in differential geometry.

Contribution

It offers detailed proofs and clarifications of Sobolev embedding and Fredholm theorems on non-compact manifolds, including conifolds and cylinders, with applications to harmonic functions.

Findings

Weighted Sobolev embedding theorems established

Fredholm properties of elliptic operators analyzed

Spaces of harmonic functions on conifolds characterized

Abstract

This paper is a self-contained presentation of certain aspects of the theory of weighted Sobolev spaces and elliptic operators on non-compact Riemannian manifolds. Specifically, we discuss (i) the standard and weighted Sobolev Embedding Theorems for general manifolds and (ii) Fredholm results for elliptic operators on manifolds with a finite number of ends modelled either on cones ("conifolds") or on cylinders. As an application of these results we present a detailed analysis of certain spaces of harmonic functions on conifolds. Some of the results presented here are of course well-known. Some others are probably known or self-evident to the experts. However, the current literature is not always easy to understand and is often sketchy, apparently not covering some aspects and consequences of the general theory which are useful in applications. In particular, in recent years results of this type have played an increasing role in Differential Geometry. The goal of this paper is thus to fill certain gaps in the current literature and to make these results available to a wider audience. The paper is also meant as a companion paper to the author's forthcoming articles [8],[9]. From this point of view it is still incomplete: future versions of this paper will incorporate more material, giving particular attention to uniform elliptic estimates for certain parametric connect sum constructions.