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We give a short and simple polynomial estimate of the norm of weighted dyadic shift on metric space with geometric doubling, which is linear in the norm of the weight. Combined with the existence of special probability space of dyadic lattices built in A. Reznikov, A. Volberg, "Random "dyadic" lattice in geometrically doubling metric space and $A_2$ conjecture", arXiv:1103.5246, and with decomposition of Calder\'on-Zygmund operators to dyadic shifts from Hyt\"onen's "The sharp weighted bound for general Calder\'on-Zygmund operators", arXiv:1007.4330 (and later T. Hyt\"onen, C. P\'erez, S. Treil, A. Volberg, "A sharp estimated of weighted dyadic shifts that gives the proof of $A_2$ conjecture", arXiv 1010.0755), we will be able to have a linear (in the norm of weight) estimate of an arbitrary Calder\'on-Zygmund operator on a metric space with geometric doubling. This will be published separately.

A Simple Sharp Weighted Estimate of the Dyadic Shifts on Metric Spaces with Geometric Doubling