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 M̄ q of Fefferman and Stein defined by M̄ q g(x) = (∑i=1∞(M f i (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 λ}) ≤ C/λ ∫ Rn |f(x)| q M w(x) dx, where C is a. constant independent of λ. © 2000 American Mathematical Society.