Abstract
Under appropriate conditions on Young’s functions Φ 1 and Φ 2, we give necessary and sufficient conditions in order that weighted integral inequalities hold for Doob’s maximal operator M on martingale Orlicz setting. When Φ 1 = t p and Φ 2 = t q, the inequalities revert to the ones of strong or weak (p, q)-type on martingale space.
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References
Gallardo D. Weighted weak type integral inequalities for the Hardy-Littlewood maximal operator. Israel J Math, 1989, 67: 95–108
Garcia-Cuerva J, Rubio de Francia J L. Weighted Norm Inequalities and Related Topics. Amsterdam: Elsevier Science Pub Co, 1985
Jawerth B. Weighted inequalities for maximal operators: linearization, localization and factorization. Amer J Math, 1986, 108: 361–414
Kikuchi M. On weighted weak type maxiaml inequalities for martingales. Math Inequal Appl, 2003, 6: 163–175
Lai Q S. Two weight mixed Φ-inequalities for the Hardy operator and the Hardy-Littlewood maximal operator. J London Math Soc, 1992, 46: 301–318
Lai Q S. Two weight Φ-inequalities for the Hardy operator, Hardy-Littlewood maximal operator, and fractional integrals. Proc Amer Math Soc, 1993, 118: 129–142
Long R L. Martingale Spaces and Inequalities. Beijing: Peking University Press, 1993
Long R L, Peng L Z. Two weighted maximal (p, q) inequalities in martingale setting (in Chinese). Acta Math Sinica, 1986, 29: 253–258
Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function. Trans Amer Math Soc, 1972, 165: 207–226
Rao M M, Ren Z D. Theory of Orlicz Spaces. New York: Marcel Dekker Inc, 1991
Ren Y B, Hou Y L. Two-weight weak-type maximal inequalities for martingales. Acta Math Scientia, 2009, 29: 402–408
Sawyer E T. Weighted norm inequalities for fractional maximal operators. In: 1980 Seminar on Harmonic Analysis (Montreal Que, 1980), CMS Conf Proc, 1. Providence, RI: Amer Math Soc, 1981, 283–309
Sawyer E T. A characterization of a two weight norm inequality for maximal operators. Studia Math, 1982, 75: 1–11
Sun Q Y. Weighted norm inequalities on spaces of homogeneous type. Studia Math, 1992, 101: 241–251
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Chen, W., Liu, P. Weighted integral inequalities in Orlicz martingale classes. Sci. China Math. 54, 1215–1224 (2011). https://doi.org/10.1007/s11425-011-4208-7
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DOI: https://doi.org/10.1007/s11425-011-4208-7