Geometric characterization of Ahlfors regular spaces in terms of dyadic cubes related to wavelets with its applications to equivalences of Lipschitz spaces

Assume that $(X,d,\mu )$ is a space of homogeneous type introduced by R. R. Coifman and G. Weiss. In this article, the authors establish a geometric characterization of Ahlfors regular spaces via the dyadic cubes constructed by T. Hytönen and A. Kairema. As applications, the authors show that Lipschitz spaces defined via the quasi-metric under consideration and Lipschitz spaces defined via the measure under consideration coincide with equivalent norms if and only if $X$ is an Ahlfors regular space. Moreover, the authors also prove that Lipschitz spaces defined via the quasi-metric under consideration and Campanato spaces defined via balls coincide with equivalent norms if and only if $X$ is an Ahlfors regular space.