Sharp weighted inequalities for the vector-valued maximal function
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- by Carlos Pérez PDF
- Trans. Amer. Math. Soc. 352 (2000), 3265-3288 Request permission
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 $\lambda$.References
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Additional Information
- Carlos Pérez
- Affiliation: Departmento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
- Email: carlos.perez@uam.es
- Received by editor(s): May 19, 1997
- Published electronically: November 18, 1999
- Additional Notes: This work was partially supported by DGICYT grant PB940192, Spain
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 3265-3288
- MSC (1991): Primary 42B20, 42B25, 42B15
- DOI: https://doi.org/10.1090/S0002-9947-99-02573-8
- MathSciNet review: 1695034