The proof of $A_2$ conjecture in a geometrically doubling metric space

TL;DR

This paper proves the $A_2$ conjecture within geometrically doubling metric spaces by constructing dyadic lattices, decomposing operators into dyadic shifts, and establishing estimates for these shifts.

Contribution

It introduces a method to prove the $A_2$ conjecture in GDMS using dyadic lattices and shift decompositions, extending previous Euclidean results.

Findings

Proof of the $A_2$ conjecture in GDMS

Construction of random dyadic lattices in metric spaces

Estimates for dyadic shifts adapted to metric setting

Abstract

We give a proof of the $A_2$ conjecture in geometrically doubling metric spaces (GDMS), i.e. a metric space where one can fit not more than a fixed amount of disjoint balls of radius $r$ in a ball of radius $2r$. Our proof consists of three main parts: a construction of a random "dyadic" lattice in a metric space; a clever averaging trick from [3], which decomposes a "hard" part of a Calderon-Zygmund operator into dyadic shifts (adjusted to metric setting); and the estimates for these dyadic shifts, made in [16] and later in [19].