  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 the
contribution 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 an
inequality of the author and 6 other collaborators. As a corollary, the same
inequality holds for all Calderon-Zygmund operators in the convex hull of Haar
shifts of a bounded complexity, of which the canonical example is the Hilbert
transform. We conjecture that the same inequality holds for all
Calderon-Zygmund operators.
