Systems of dyadic cubes in a doubling metric space

TL;DR

This paper extends the concept of adjacent dyadic cube systems from Euclidean spaces to general metric spaces with doubling measures, enabling advanced harmonic analysis techniques in broader contexts.

Contribution

It introduces a new deterministic construction of multiple adjacent dyadic systems in metric spaces of homogeneous type, expanding analytical tools beyond Euclidean settings.

Findings

Constructed boundedly many adjacent dyadic systems with covering properties

Streamlined a random construction method for dyadic systems

Applied constructions to weighted inequalities and BMO spaces

Abstract

A number of recent results in Euclidean Harmonic Analysis have exploited several adjacent systems of dyadic cubes, instead of just one fixed system. In this paper, we extend such constructions to general spaces of homogeneous type, making these tools available for Analysis on metric spaces. The results include a new (non-random) construction of boundedly many adjacent dyadic systems with useful covering properties, and a streamlined version of the random construction recently devised by H. Martikainen and the first author. We illustrate the usefulness of these constructions with applications to weighted inequalities and the BMO space; further applications will appear in forthcoming work.