Sharp Weighted Inequalities for the Vector–valued Maximal Function

We prove in this paper some sharp weighted inequalities for the vector–valued maximal function Mq of Fefferman and Stein defined by Mqf(x) = ( ∞ ∑ i=1 (Mfi(x)) q )1/q , where M is the Hardy–Littlewood maximal function. As a consequence we derive the main result establishing that in the range 1 < q < p < ∞ there exists a constant C such that ∫ Rn Mqf(x) p w(x)dx ≤ C ∫ Rn |f(x)|qM [ p q ]+1 w(x)dx. Furthermore the result is sharp since M [ p q ]+1 cannot be replaced by M [ p q ] . We also show the following endpoint estimate w({x ∈ R : Mqf(x) > λ}) ≤ C λ ∫ Rn |f(x)|qMw(x)dx, where C is a constant independent of λ. 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 M q be the vector–valued maximal operator defined by