Function Spaces on Singular Manifolds

TL;DR

This paper extends fundamental results for Sobolev and Bessel potential spaces to Riemannian manifolds with singularities, using weighted spaces to account for boundary and singularity effects, aiding PDE analysis on complex domains.

Contribution

It demonstrates that classical function space results remain valid on singular manifolds when appropriate weights are used, broadening the scope of PDE analysis tools.

Findings

Most Sobolev-Slobodeckii and Bessel potential space results hold on singular manifolds with weights.

Weighted function spaces effectively handle singularities and boundaries.

Results facilitate PDE study on piecewise smooth domains.

Abstract

It is shown that most of the well-known basic results for Sobolev-Slobodeckii and Bessel potential spaces, known to hold on bounded smooth domains in $\mathbb{R}^n$, continue to be valid on a wide class of Riemannian manifolds with singularities and boundary, provided suitable weights, which reflect the nature of the singularities, are introduced. These results are of importance for the study of partial differential equations on piece-wise smooth domains.