Anisotropic Function Spaces on Singular Manifolds

TL;DR

This paper thoroughly studies anisotropic function spaces on singular manifolds, establishing key properties and embeddings crucial for analyzing time-dependent PDEs in complex geometric settings.

Contribution

It provides a comprehensive analysis of weighted anisotropic Bessel potential, Besov, and Hölder spaces on singular manifolds with boundary, including embeddings, trace theorems, and boundary condition characterizations.

Findings

Derived Sobolev-type embedding results

Established sharp trace theorems for boundary spaces

Analyzed point-wise multiplier properties and interpolation

Abstract

A rather complete investigation of anisotropic Bessel potential, Besov, and H\"older spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends' is determined by a singularity function which leads naturally to the study of weighted function spaces. Besides of the derivation of Sobolev-type embedding results, sharp trace theorems, point-wise multiplier properties, and interpolation characterizations particular emphasize is put on spaces distinguished by boundary conditions. This work is the fundament for the analysis of time-dependent partial differential equations on singular manifolds.