Random "dyadic" lattice in geometrically doubling metric space and $A_2$ conjecture

Abstract

Recently three proofs of the $A_2$-conjecture were obtained. All of them are "glued" to euclidian space and a special choice of one random dyadic lattice. We build a random "dyadic" lattice in any doubling metric space which have properties that are enough to prove the $A_2$-conjecture in these spaces.