Function Spaces on Uniformly Regular and Singular Riemannian Manifolds

TL;DR

This paper extends the theory of function spaces like Sobolev, Besov, and Bessel potential spaces to Riemannian manifolds with boundary, including those with singularities, and demonstrates their application to maximal regularity in parabolic problems.

Contribution

It establishes the validity of fundamental properties of these function spaces on singular and bounded geometry manifolds, introducing Kondratiev-type weighted spaces for singular cases.

Findings

Properties of Sobolev, Besov, and Bessel spaces hold on manifolds with boundary and singularities.

Maximal regularity results are proven for linear parabolic problems on singular manifolds.

Weighted Kondratiev spaces are effective for analysis on manifolds with singularities.

Abstract

This paper shows that the basic properties of Sobolev, Besov, and Bessel potential spaces are valid on Riemannian manifolds with boundary, which either have bounded geometry or posses singularities. In the latter case the appropriate setting is that of Kondratiev-type weighted spaces. The importance and usefulness of our results are indicated by a demonstration of a maximal regularity result for a linear parabolic initial value problem on singular manifolds.