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

We show that for 1 < p < ∞, weight w ∈ Ap, and any L2-bounded Calderón-Zygmund operator T, there is a constant CT,p such that the weak- and strong-type inequalities $${\left\| {{T_\natural}f} \right\|_{{L^{p,\infty }}(w)}} \le {C_{T,p}}{\left\| w \right\|_{{A_p}}}{\left\| f \right\|_{{L^p}(w )}}$$$${\left\| {{T_\natural}f} \right\|_{{L^p}(w)}} \le {C_{T,p}}\left\| w \right\|_{{A_p}}^{\max \{ 1,{{(p - 1)}^{ - 1}}}{\left\| f \right\|_{{L^p}(w)}}$$ hold, where T♮ denotes the maximal truncations of T and $${\left\| w \right\|_{{A_p}}}$$ denotes the Muckenhoupt Ap characteristic of w. These estimates are not improvable in the power of $${\left\| w \right\|_{{A_p}}}$$. 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♮ and sufficient conditions for two-weight inequalities in Lp for T♮. Our proof does not rely upon extrapolation.