Sphericalization and flattening with their applications in quasimetric measure spaces

The main purpose of the note is to explore the invariant properties of sphericalization and flattening and their applications in quasi-metric spaces. We show that sphericalization and flattening procedures on a quasimetric spaces preserving properties such as Ahlfors regular and doubling property. By using these properties, we generalize a recent result in \cite{WZ}. We also show that the Loewner condition can be preserved under quasim"obius mapping between two $Q$-Ahlfors regular spaces. Finally, we prove that the $Q$-regularity of $Q$-dimensional Hausdorff measure of Bourdon metric are coincided with Hausdorff measure of Hamenst"adt metric defined on the boundary at infinity of a Gromov hyperbolic space.