Locally biHölder continuous maps and their induced embeddings between Besov spaces

We introduce a class of homeomorphisms between metric spaces, which are locally biHolder continuous maps. Then an embedding result between Besov spaces induced by locally biHolder continuous maps between Ahlfors regular spaces is established, which extends the corresponding result of Bjorn, Bjorn, Gill, and Shanmugalingam. 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 map between uniformly perfect spaces is locally biHolder continuous.