The proof of $A_2$ conjecture in a geometrically doubling metric space Fedor NazarovAlexander ReznikovAlexander Volberg 42B20Calder\\\'on--Zygmund operatorsweightsBellman functionrandom dyadic shifts We give a proof of the $A_2$ conjecture in geometrically doubling metric spaces (GDMS), that is, a metric space where one can fit no 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 [T.\:P. Hyt\"onen, \textit{The sharp weighted bound for general Calder\'on-Zygmund operators}, Ann. of Math. (2) \textbf{175} (2012), no. 3, 1473--1506], which decomposes a "hard" part of a Calder\'on-Zygmund operator into dyadic shifts (adjusted to metric setting); and the estimates for these dyadic shifts, made in [F. Nazarov and A. Volberg, \textit{A simple sharp weighted estimate of the dyadic shifts on metric spaces with geometric doubling}, Int. Math. Res. Not. IMRN \textbf{16} (2013), 3771--3789] and later in [S. Treil, \textit{Sharp $A_2$ estimates of Haar shifts via Bellman function}, available at http://arxiv.org/abs/arXiv:1105.2252v1]. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.5098 10.1512/iumj.2013.62.5098 en Indiana Univ. Math. J. 62 (2013) 1503 - 1533 state-of-the-art mathematics http://iumj.org/access/