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Sharp weighted inequalities for the vector-valued maximal function

Author(s): Carlos Pérez
Journal: Trans. Amer. Math. Soc. 352 (2000), 3265-3288.
MSC (1991): Primary 42B20, 42B25, 42B15
Posted: November 18, 1999
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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 $M$ 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

\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 $C$ is a constant independent of $\lambda$.

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Additional Information:

Carlos Pérez
Affiliation: Departmento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: carlos.perez@uam.es

DOI: 10.1090/S0002-9947-99-02573-8
PII: S 0002-9947(99)02573-8
Received by editor(s): May 19, 1997
Posted: November 18, 1999
Additional Notes: This work was partially supported by DGICYT grant PB940192, Spain
Copyright of article: Copyright 2000, American Mathematical Society

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