Borderline weak-type estimates for singular integrals and square functions

For any Calderón-Zygmund operator T, any weight w, and α >1, the operator T is bounded as a map from L¹ (M_{L log log L (log log log L)}^αw) into weak-L¹(w). The interest in questions of this type goes back to the beginnings of the weighted theory, with prior results, due to Coifman-Fefferman, Pérez, and Hytönen-Pérez, on the L (log L)^ε scale. Also, for square functions S f, and weights w ∈ A_p, the norm of S from L^p(w) to weak-L^p(w), 2≤ p < ∞, is bounded by [w]_Ap^1/2 (1+log [w]_A∞)^1/2, which is a sharp estimate.