Locally biH\"{o}lder continuous mappings and their induced embeddings between Besov spaces

TL;DR

This paper studies locally biHölder continuous homeomorphisms between metric spaces and their impact on embeddings of Besov spaces, extending previous results and introducing new geometric conditions for such mappings.

Contribution

It extends known embedding results between Besov spaces via locally biHölder continuous mappings and introduces the concept of uniform boundedness for characterizing quasisymmetric mappings.

Findings

Established a new embedding result for Besov spaces induced by locally biHölder continuous mappings.

Constructed an example demonstrating the generality of the new embedding result.

Introduced the uniform boundedness condition to characterize when quasisymmetric mappings are locally biHölder continuous.

Abstract

In this paper, we introduce a class of homeomorphisms between metric spaces, which are locally biH\"{o}lder continuous mappings. Then an embedding result between Besov spaces induced by locally biH\"{o}lder continuous mappings between Ahlfors regular spaces is established, which extends the corresponding result of Bj\"{o}rn-Bj\"{o}rn-Gill-Shanmugalingam (J. Reine Angew. Math. 725: 63-114, 2017). Furthermore, an example is constructed to show that our embedding result is more general. We also introduce a geometric condition, named as uniform boundedness, to characterize when a quasisymmetric mapping between uniformly perfect spaces is locally biH\"{o}lder continuous.