Sharp weighted inequalities for the vector-valued maximal function

Abstract

We prove in this paper some sharp weighted inequalities for the vector-valued maximal function Mq of Fefferman and Stein defined by 00 \llq Mqf (X) (Mfi (o q where M is the Hardy-Littlewood maximal function. As a consequence we derive the main result establishing that in the range 1 A}) <-| Jf(X)Jq Mw(x)dx, A Rn where C is a constant independent of A. 1. MOTIVATION AND DESCRIPTION OF THE MAIN RESULTS The purpose of this paper is to obtain some sharp weighted inequalities for the vector-valued maximal function Mq which are not within the scope of the standard Ap theory for vector-valued singular integrals as can be found in [RRT]. We start with a review of some of the classical estimates and then we shall state the main results. 1.1. Background. Let M be the Hardy-Littlewood maximal function and let Mq be the vector-valued maximal operator defined by Mq f (x) E(M fi (x) )q This nonlinear operator was introduced by C. Fefferman and E. M. Stein in [FS] as a generalization of both the (scalar) maximal function M and the classical integral of Marcinkiewicz and since then it has played an important role in the development of modern Harmonic Analysis. We recall the two basic estimates obtained in [FS] for 1 < q < oo: Received by the editors May 19, 1997. 1991 Mathematics Subject Classification. Primary 42B20, 42B25, 42B15. This work was partially supported by DGICYT grant PB940192, Spain. (@)2000 American Mathematical Society 3265 This content downloaded from 207.46.13.102 on Sat, 30 Jul 2016 04:48:12 UTC All use subject to http://about.jstor.org/terms