Abstract
Let T be an arbitrary L^2 bounded Calderon--Zygmund operator, and T_# its maximal truncated version. Then T_# satisfies the following bound for all 1<p<\infty and all weights w\in A_p: \|T_# \|_{L^p(w)} << [w]_{A_p}^{1/p} {[w]_{A_infty}^{1/p'}+[w^{1-p'}]_{A_infty}^{1/p}}.
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