Systems of Dyadic Cubes of Complete, Doubling, Uniformly Perfect Metric Spaces without Detours

TL;DR

This paper constructs dyadic cube systems in complete, doubling, uniformly perfect metric spaces, enabling chains of cubes to connect any two points with diameters comparable to their distance, advancing harmonic analysis tools.

Contribution

It introduces a new construction of dyadic cubes in specific metric spaces, facilitating analysis and geometric applications without detours.

Findings

Constructed dyadic cube systems with chain properties

Enabled analysis tools in complete, doubling, uniformly perfect spaces

Applied to potential analysis and geometric research

Abstract

Systems of dyadic cubes are the basic tools of harmonic analysis and geometry, and this notion had 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. We also give an application of our construction to previous research of potential analysis and geometry of metric spaces.