A simple proof of the sharp weighted estimate for Calderon-Zygmund operators on homogeneous spaces

Here we show that Lerner's method of local mean oscillation gives a simple proof of the $A_2$ conjecture for spaces of homogeneous type: that is, the linear dependence on the $A_2$ norm for weighted $L^2$ Calderon-Zygmund operator estimates. In the Euclidean case, the result is due to Hyt\"{o}nen, and for geometrically doubling spaces, Nazarov, Rezinikov, and Volberg obtained the linear bound.