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

Abstract

The classical Rubio de Francia extrapolation result asserts that if an operator T:Lp0(u)→Lp0,∞(u) is bounded for some p0>1 and every u∈Ap0, then, for every 1<p<∞ and every u∈Ap, T:Lp(u)→Lp,∞(u) is bounded. However, there are examples showing that it is not possible to extrapolate to the end-point p=1. In this paper we shall prove that there exists a class of weights, slightly larger than Ap, with the following property: If an operator T:Lp0,1(u)→Lp0,∞(u) is bounded, for some p0>1 and every u in this class then, for every u∈A1,(1)T is of restricted weak-type (1,1);(2)for every ε>0,T:L(log⁡L)ε(u)⟶Lloc1,∞(u). T is of restricted weak-type (1,1); for every ε>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) estimate with respect to every u∈A1.