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Local $Tb$ theorems with $L^p$ type testing conditions, which are not scaleinvariant, have been studied widely in the case of the Lebesgue measure. Untilvery recently, local $Tb$ theorems in the non-homogeneous case had only beenproved assuming scale invariant ($L^{\infty}$ or BMO) testing conditions. Thecombination of non-scale-invariance and general measures is a delicate issue.In a previous paper we overcame this obstacle in the model case of squarefunctions defined using general measures. In this paper we finally tackle thevery demanding case of Calder\'on-Zygmund operators. That is, we prove anon-homogeneous local $Tb$ theorem with $L^2$ type testing conditions for allCalder\'on-Zygmund operators. In doing so we prove general twisted martingaletransform inequalities which turn out to be subtle in our general framework.

Local $Tb$ theorem with $L^2$ testing conditions and general measures: Calder\'on-Zygmund operators