Optimal factorization of muckenhoupt weights

Peter Jones' theorem on the factorization of A(p) weights is sharpened for weights with bounds near 1, allowing the factorization to be performed continuously near the limiting, unweighted case. When 1 < p < infinity and w is an A(p) weight with bound A(p)(w) = 1 + epsilon, it is shown that there exist A(1) weights u, v such that both the formula w = uv(1-p) and the estimates A(1)(u), A(1)(v) = 1 + O (root epsilon) hold. The square root in these estimates is also proven to be the correct asymptotic power as epsilon --> 0.