Sharp weighted inequalities for the vector-valued maximal function

We prove in this paper some sharp weighted inequalities for the vector-valued maximal function (M) over bar(q) of Fefferman and Stein defined by [GRAPHICS] where M is the Hardy-Littlewood maximal function. As a consequence we derive the main result establishing that in the range 1 < q < p < infinity there exists a constant C such that integral(Rn) (M) over bar(q)f(x)(p) w(x)dx less than or equal to C integral(Rn) \f(x)\(p)(q) M[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 is an element of R-n : (M) over bar(q)f(x) > lambda}) less than or equal to C/lambda integral(Rn) \f(x)\(q) Mw(x)dx, where C is a constant independent of lambda.