Publishing house / Journals and Serials / Colloquium Mathematicum / All issues

Abstract

Systems of dyadic cubes are the basic tools of harmonic analysis and geometry, and this notion has been extended to general metric spaces. In this paper, we construct systems of dyadic cubes of complete, doubling, uniformly perfect metric spaces, such that for any two points in the metric space, there exists a chain of three cubes whose diameters are comparable to the distance of the points. As an application, we show that our construction can be applied to a framework of tree-like partitions used in previous results on potential-theoretic evaluation of the Ahlfors regular conformal dimension of metric spaces. Furthermore, we show that for a complete metric space without isolated points, the existence of such a partition is equivalent to the Ahlfors regular conformal dimension of the space being finite.