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 Mqf(x) = X∞ i=1 (M fi(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 Z Rn Mqf(x)p w(x)dx ≤ C Z Rn |f(x)|p q M[ p q ]+1w(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 ∈ Rn : Mqf(x) > λ}) ≤ C λ Z Rn |f(x)|q Mw(x)dx, where C is a constant independent of λ.