Relations between Kondratiev spaces and refined localization Triebel-Lizorkin spaces

TL;DR

This paper explores the relationship between Kondratiev spaces and Triebel-Lizorkin refined localization spaces, improving an embedding result that impacts adaptive approximation convergence rates.

Contribution

It provides a significant enhancement of an existing embedding between Kondratiev and Triebel-Lizorkin spaces, utilizing a new characterization of refined localization spaces.

Findings

Improved embedding results for Kondratiev and Triebel-Lizorkin spaces

Enhanced understanding of convergence rates in adaptive schemes

Application of new characterizations to refine space relations

Abstract

We investigate the close relation between certain weighted Sobolev spaces (Kondratiev spaces) and refined localization spaces from introduced by Triebel [39,40]. In particular, using a characterization for refined localization spaces from Scharf [32], we considerably improve an embedding from Hansen [17]. This embedding is of special interest in connection with convergence rates for adaptive approximation schemes.