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 $\overline M_q$ of Fefferman and Stein defined by \[ \overline M_qf(x)=\left (\sum _{i=1}^{\infty }(Mf_i(x))^{q}\right )^{1/q},\] where $M$ is the Hardy–Littlewood maximal function. As a consequence we derive the main result establishing that in the range $1<q<p<\infty$ there exists a constant $C$ such that \[ \int _{\mathbf {R}^{n}}\overline M_qf(x)^p w(x)dx\le C \int _{\mathbf {R}^n}|f(x)|^{p}_{q} M^{[\frac pq]+1}w(x) dx.\] Furthermore the result is sharp since $M^{[\frac pq]+1}$ cannot be replaced by $M^{[\frac pq]}$. We also show the following endpoint estimate \[ w(\{x\in \mathbf {R}^n:\overline M_qf(x)>\lambda \}) \le \frac C\lambda \int _{\mathbf {R}^n} |f(x)|_q Mw(x)dx,\] where $C$ is a constant independent of $\lambda$.