Borderline weak-type estimates for singular integrals and square functions

domingo-salazar, carlos; lacey, michael; rey, guillermo;

For any Calder´on–Zygmund operator T, any weight w, and α > 1, the operator T is bounded as a map from L1(ML 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, P´erez, and Hyt¨onen–P´erez, on the L(log L) scale. Also, for square functions Sf,and weights w ∈ Ap, the norm of S from Lp(w) to weak-Lp(w), 2  p < ∞, is bounded by[w]1/2
Ap (1 + log[w]A∞)1/2, which is a sharp estimate.