Locally biHölder continuous mappings and their induced embeddings between Besov spaces

In this paper, we introduce a class of homeomorphisms between metric spaces, which are locally biHölder continuous mappings. Then an embedding result between Besov spaces induced by locally biHölder continuous mappings between Ahlfors regular spaces is established, which extends the corresponding result of Björn-Björn-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 biHölder continuous.