Borderline weak-type estimates for singular integrals and square functions

For any Calderon-Zygmund operator T, any weight w, and alpha > 1, the operator T is bounded as a map from L-1 (M-L log log L(log log log L) alpha w) into weak-L-1 (w). The interest in questions of this type goes back to the beginnings of the weighted theory, with prior results, due to Coifman-Fefferman, Perez, and Hytonen-Perez, on the L(log L)(epsilon) scale. Also, for square functions Sf, and weights w is an element of A(p), the norm of S from L-p (w) to weak-L-p (w), 2 <= p < infinity, is bounded by [w](Ap)(1/2)(1 + log[w](A infinity))(1/2), which is a sharp estimate.