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Continuing a theme of Lerner and Hytonen-Perez, we establish an L^p(w)inequality for a Haar shift operator of bounded complexity, that quantifies thecontribution of the A_infty characteristic of the weight to the L^p norm. Here,1<p<\infty. The Hytonen-Perez inequality is only for p=2, and we improve aninequality of the author and 6 other collaborators. As a corollary, the sameinequality holds for all Calderon-Zygmund operators in the convex hull of Haarshifts of a bounded complexity, of which the canonical example is the Hilberttransform. We conjecture that the same inequality holds for allCalderon-Zygmund operators.

An A_p --A_infty inequality for the Hilbert Transform