Weighted Weak-type (1, 1) Estimates via Rubio De Francia Extrapolation

The classical Rubio de Francia extrapolation result asserts that if an operator T : L0(u) → Lp0,∞(u) is bounded for some p0 > 1 and every u ∈ Ap0 , then, for every 1 < p < ∞ and every u ∈ Ap, T : L(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 : L0(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(logL)(u) −→ L loc (u). Moreover, for a big class of operators, including Calderón-Zygmund maximal operators, g-functions, the intrinsic square function, and the Haar shift operators, we obtain a weak-type (1, 1) estimate with respect to every u ∈ A1.