Local Tb theorem with L², testing conditions and general Measures: Calderón-zygmund operators

Local Tb theorems with L² type testing conditions have been studied widely in the case of the Lebesgue measure. Such conditions are tied to the scale of the given test function's supporting cube. Until very recently, local Tb theorems in the non-homogeneous case had only been proved assuming scale invariant (L^∞, or BMO) testing conditions. Moving past such strong assumptions in non-homogeneous analysis is a key problem. In a previous paper we overcame this obstacle in the model case of square functions defined using general measures. In this paper we finally tackle the very demanding case of Calderón-Zygmund operators. That is, we prove a non-homogeneous local Tb theorem with L², type testing conditions for all Calderón-Zygmund operators. In doing so we prove general twisted martingale transform inequalities which turn out to be subtle in our general framework.