Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on A _p weighted spaces

We show that for 1 < p < ∞, weight w ∈ A _p, and any L ²-bounded Calderón-Zygmund operator T, there is a constant C _T,p such that the weak- and strong-type inequalities hold, where T _{{music natural sign}} denotes the maximal truncations of T and {double pipe}w{double pipe} _Ap denotes the Muckenhoupt A _p characteristic of w. These estimates are not improvable in the power of {double pipe}w{double pipe} _Ap. Our argument follows the outlines of those of Lacey-Petermichl-Reguera (Math. Ann. 2010) and Hytönen-Pérez-Treil-Volberg (arXiv, 2010) and contains new ingredients, including a weak-type estimate for certain duals of T _{{music natural sign}} and sufficient conditions for two-weight inequalities in L ^p for T _{{music natural sign}}. Our proof does not rely upon extrapolation.