A note on dyadic coverings and nondoubling Calder\'on-Zygmund theory

TL;DR

This paper develops an optimal family of dyadic filtrations in Euclidean space, enabling a new nondoubling Calderón-Zygmund decomposition that simplifies previous approaches and extends to general metric spaces.

Contribution

It constructs an optimal dyadic covering in R^n, improving previous methods, and introduces a simplified nondoubling Calderón-Zygmund decomposition applicable to metric spaces.

Findings

Constructed n+1 dyadic filtrations covering Euclidean balls efficiently.

Provided a dyadic nondoubling Calderón-Zygmund decomposition avoiding Besicovitch coverings.

Extended the decomposition to upper doubling metric spaces.

Abstract

We construct a family of n+1 dyadic filtrations in R^n, so that every Euclidean ball B is contained in some cube Q of our family satisfying diam(Q) \le c_n diam(B) for some dimensional constant c_n. Our dyadic covering is optimal on the number of filtrations and improves previous results of Christ and Garnett/Jones by extending a construction of Mei for the n-torus. Based on this covering and motivated by applications to matrix-valued functions, we provide a dyadic nondoubling Calder\'on-Zygmund decomposition which avoids Besicovitch type coverings in Tolsa's decomposition. We also use a recent result of Hyt\"onen and Kairema to extend our dyadic nondoubling decomposition to the more general setting of upper doubling metric spaces.