A weak to strong type T1 theorem for general smooth Calderón-Zygmund operators with doubling weights, II

Abstract: We consider the weak to strong type problem for two weight norm inequalities for Calder\'on-Zygmund operators with doubling weights. We show that if a Calder\'on-Zygmund operator T is weak type (2,2) with doubling weights, then it is strong type (2,2) if and only if the dual cube testing condition for T^{*} holds, alternatively if and only if the dual cancellation condition of Stein holds. The testing condition can be taken with respect to either cubes or balls, and more generally, this is extended to a weak form of Tb theorem. Finally, we show that for all pairs of locally finite positive Borel measures, and all Stein elliptic Calder\'on-Zygmund operators T, the weak type (2,2) inequalities for T and and its associated maximal truncations operator T_{} are equivalent. Thus the characterization of weak type for T_{} in [LaSaUr1] applies to T as well.