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

作者：Carro Maria J^*; Grafakos Loukas; Soria Javier

来源：Journal of Functional Analysis, 2015, 269(5): 1203-1233.

DOI：10.1016/j.jfa.2015.06.005

• Rubio de Francia extrapolation
• A(p) weights
• Hardy-Littlewood maximal function

The classical Rubio de Francia extrapolation result asserts that if an operator T : L-P0 (u) -> L-P0,L-infinity(u) is bounded for some p(0) > 1 and every U is an element of A(P0), then, for every 1 < p < infinity and every u is an element of A(p), T: L-P(u) -> L-P,L-infinity(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 operatorT : L-P0,L-1 (u) -> L-P0,L-infinity(u) is bounded, for some p(0) > 1 and every u in this class then, for every u is an element of A(1),
(1) T is of restricted weak-type (1,1);
(2) for every epsilon > 0, T: L(log L)(epsilon)(u) -> L-loc(1,infinity)(u).
Moreover, for a big class of operators, including Calderon-Zygmund maximal operators, g-functions, and the intrinsic square function, we obtain a weak-type (1,1) estimate with respect to every u is an element of A(1).

• 出版日期2015-9-1