×

The Fefferman-Stein type inequalities for strong fractional maximal operators. (English) Zbl 1404.42039

Summary: We prove the Fefferman-Stein type inequalities for strong fractional maximal operators by additional compositions of certain maximal operators instead of using the strong Muckenhoupt weight. With an arbitrary weight, in \({{\mathbb {R}}}^2\), we establish an endpoint estimate and in \({{\mathbb {R}}}^n\), \(n\geq 2\), we give a weak (\(p\), \(p\)) type estimate for \(p>1\). We also investigate the case \(p=1\) in higher dimensions.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cruz-Uribe, D., SFO, New proofs of two-weight norm inequalities for the maximal operator, Georgian Math. J., 7, 33-42, (2000) · Zbl 0987.42019
[2] Fava, N., Weak type inequalities for product operators, Studia Math., 42, 271-288, (1972) · Zbl 0237.47006 · doi:10.4064/sm-42-3-271-288
[3] Fefferman, C.; Stein, EM, Some maximal inequalities, Am. J. math., 93, 107-115, (1971) · Zbl 0222.26019 · doi:10.2307/2373450
[4] Fefferman, R.; Pipher, J., A covering lemma for rectangles in \({{\mathbb{R}}}^n\), Proc. Am. Math. Soc., 133, 3235-3241, (2005) · Zbl 1084.42015 · doi:10.1090/S0002-9939-05-07902-5
[5] Garcia-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland, Math. Stud., vol.116 (1985) · Zbl 0578.46046
[6] Jawerth, B.; Torchinsky, A., The strong maximal function with respect to measures, Studia Math., 80, 261-285, (1984) · Zbl 0565.42008 · doi:10.4064/sm-80-3-261-285
[7] Jessen, B.; Marcinkiewicz, J.; Zygmund, A., Note on the differentiability of multiple integrals, Fundam. Math., 25, 217-234, (1935) · Zbl 0012.05901 · doi:10.4064/fm-25-1-217-234
[8] Kokilashvili, V.; Meskhi, A., Two-weight estimates for strong fractional maximal functions and potentials with multiple kernels, J. Korean Math. Soc., 46, 523-550, (2009) · Zbl 1201.42013 · doi:10.4134/JKMS.2009.46.3.523
[9] Lin, K.-C.: Ph.D. University of California, Los Angeles 1984 United States. Dissertation: Harmonic Analysis on the Bidisc
[10] Luque, T.; Parissis, I., The endpoint Fefferman-Stein inequality for the strong maximal function, J. Funct. Anal., 266, 199-212, (2014) · Zbl 1310.42011 · doi:10.1016/j.jfa.2013.09.028
[11] Mitsis, T., The weighted weak type inequality for the strong maximal function, J. Fourier Anal. Appl., 12, 645-652, (2006) · Zbl 1114.42007 · doi:10.1007/s00041-005-5060-3
[12] Pérez, C., A remark on weighted inequalities for general maximal operators, Proc. Am. Math. Soc., 119, 1121-1126, (1993) · Zbl 0810.42008 · doi:10.2307/2159974
[13] Saito, H.; Tanaka, H., The Fefferman-Stein type inequalities for strong and directional maximal operators in the plane, Can. Math. Bull., 61, 191-200, (2018) · Zbl 1388.42058 · doi:10.4153/CMB-2017-024-x
[14] Sawyer, E., Weighted norm inequalities for fractional maximal operators, Proc. C. M. S., 1, 283-309, (1981) · Zbl 0546.42018
[15] Sawyer, E., Wang, Z.: Weighted inequalities for product fractional integrals. Preprint arXiv:1702.03870
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.