×

A characterization of some weighted inequalities for the vector-valued weighted maximal function. (English) Zbl 1228.42026

The author obtains necessary and sufficient conditions of weighted weak and strong type norm inequalities for vector-valued weighted maximal functions. More precisely, let \(f=\{f_k\}_1^\infty\) be a sequence of locally integrable functions on \(\mathbb R^n\), \(|f(x)|_r= (\sum_1^\infty| f(x)|^r)\) and \(M\omega f(x)= \sup_{x\in Q}\int_Q|f(y)|\omega(y)\,dy\). The author proves that
(1)
There is constant \(C_{r,p}\) such that
\[ u\big(\{x\in\mathbb R^n:| M\omega f(x)|_r>\lambda\}\big)\leq C_{r,p} \lambda^{-p} \int_{\mathbb R^n}|f(y)|_r^p u(y)\,dy \]
if and only if \(u\in A_{p}(\omega)\).
(2)
There is a constant \(C_{r,p}\) such that
\[ \int_{\mathbb R^n}|M_\omega f(y)|_r^p u(y)\,dy\leq C_{r,p} \int_{\mathbb R^n}|f(y)|_r^p u(y)\,dy \]
if and only if \(u\in A_{p}(\omega)\).
The case \(\omega(x)=1\) was proved by K. F. Andersen and R. T. John [Stud. Math. 69, 19–31 (1980; Zbl 0448.42016)].

MSC:

42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 0448.42016
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Tran. Amer. Math. Soc., 165, 207–226 (1972) · Zbl 0236.26016 · doi:10.1090/S0002-9947-1972-0293384-6
[2] Luo, C.: A strong type inequality about the maximal operator M w on L p(\(\mathbb{R}\)n, udx) and its application to the Riesz potential. Acta Mathematica Sinica, Chinese Series, 42(6), 969–974 (1999) · Zbl 1009.42012
[3] Wu, C. L., Shu, L. S.: Weak type estimate about weighted maximal operator M {\(\omega\)} on L p(\(\mathbb{R}\)n, udx). Journal of Nanjing University Mathematical Biquarterly, 20(1), 16–23 (2003) · Zbl 1073.42016
[4] Andersen, K. F., John, R. T.: Weighted inequalities for vector-valued maximal function and singular integrals. Studia Math., 69, 19–31 (1980) · Zbl 0448.42016
[5] Stein, E. M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, New Jersy, 1993 · Zbl 0821.42001
[6] Zhou, M. Q.: Lecture of Harmonic Analysis (Real-Variable Methods), Beijing Univ. Press, Beijing, 1999
[7] Fefferman, C., Stein, E. M.: Some maximal inequalities. Amer. J. Math., 93, 107–115 (1971) · Zbl 0222.26019 · doi:10.2307/2373450
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.