Weighted norm inequalities for the Hardy-Littlewood maximal operator on spaces of homogeneous type

Abstract

The purpose of this note is to give an adequate Calderon-Zygmund type lemma in order to extend to the general setting of spaces of homogeneous type the Ap weighted Lp boundedness for the Hardy-Littlewood maximal operator given by M. Christ and R. Fefferman. Recently Michael Christ and Robert Fefferman gave in (1) a remarkable proof of the weighted norm inequality for the Hardy-Littlewood maximal function operator in R, \\Mf\\Lp(w) 1. In (2), A. P. Calderon proved this boundedness property for spaces such that the measure of balls is continuous as a function of the radius. In (3), R. Macias and C. Segovia extended this result to general spaces of homogeneous type (defined below) constructing an adequate quasi-distance. In both cases, the reverse Holder inequality must be extended to this general setting, while the proof given in (1) does not make use of this property and only depends on an adequate Calderon-Zygmund type lemma, the proof of which for cubes in R is very simple. The purpose of this note is to obtain a decomposition lemma which allows us to extend the proof of