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The main purpose of the note is to explore the invariant properties ofsphericalization and flattening and their applications in quasi-metric spaces.We show that sphericalization and flattening procedures on a quasimetric spacespreserving properties such as Ahlfors regular and doubling property. By usingthese properties, we generalize a recent result in \cite{WZ}. We also show thatthe 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 withHausdorff measure of Hamenst\"adt metric defined on the boundary at infinity ofa Gromov hyperbolic space.

Sphericalization and flattening with their applications in quasimetric measure spaces