Weighted weak-type (1,1)(1,1) estimates via Rubio de Francia extrapolation

The classical Rubio de Francia extrapolation result asserts that if an operator T:Lp0(u)→Lp0,∞(u)T:Lp0(u)→Lp0,∞(u) is bounded for some p0>1p0>1 and every u∈Ap0u∈Ap0, then, for every 11p0>1 and every u   in this class then, for every u∈A1u∈A1,(1)T   is of restricted weak-type (1,1)(1,1);(2)for every ε>0ε>0,T:L(log⁡L)ε(u)⟶Lloc1,∞(u).Moreover, for a big class of operators, including Calderón–Zygmund maximal operators, g  -functions, and the intrinsic square function, we obtain a weak-type (1,1)(1,1) estimate with respect to every u∈A1u∈A1.