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Sharp weighted inequalities for the vector-valued maximal function

Authors:	Carlos Pé rez

Institution:	Departmento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

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 $\begin{displaymath}\overline M_qf(x)=\left(\sum _{i=1}^{\infty}(Mf_i(x))^{q}\right)^{1/q},\end{displaymath}$ where 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 such that $\begin{displaymath}\int _{\mathbf{R}^{n}}\overline M_qf(x)^p\, w(x)dx\le C\, \int _{\mathbf{R}^n}\|f(x)\|^{p}_{q}\, M^{\frac pq]+1}w(x) dx.\end{displaymath}$ Furthermore the result is sharp since $M^{\frac pq]+1}$ cannot be replaced by $M^{\frac pq]}$ . We also show the following endpoint estimate $\begin{displaymath}w(\{x\in \mathbf{R}^n:\overline M_qf(x)>\lambda\})\,\le \frac C\lambda \int _{\mathbf{R}^n} \|f(x)\|_q\, Mw(x)dx,\end{displaymath}$ where is a constant independent of $\lambda$ .

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