For any Calderón–Zygmund operator T, any weight w, and α>1, the operator T is bounded as a map from L1false(ML0.166667emlog0.166667emlog0.166667emLfalse(log0.166667emlog0.166667emlog0.166667emLfalse)αwfalse) into weak‐L1false(wfalse). 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(logL)ϵ scale. Also, for square functions Sf, and weights w∈Ap, the norm of S from Lpfalse(wfalse) to weak‐Lpfalse(wfalse), 2⩽p<∞, is bounded by [w]Ap1/2(1+log[w]A∞)1/2, which is a sharp estimate.

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