A FOUR-WEIGHT WEAK TYPE MAXIMAL INEQUALITY FOR MARTINGALES

doi:10.1007/s10473-019-0207-5

Abstract

Abstract: In this article, some necessary and sufficient conditions are shown in order that weighted inequality of the form

holds a.e. for uniformly integrable martingales f = (f_n)n≥0 with some constant C > 0, where Φ₁, Φ₂ are Young functions, w_i (i = 1, 2, 3, 4) are weights, and f_∞ = f_n a.e. As an application, two-weight weak type maximal inequalities of martingales are considered, and particularly a new equivalence condition is presented.

Key words: weight, weak type inequality, martingale maximal operator, Young function

Yanbo REN. A FOUR-WEIGHT WEAK TYPE MAXIMAL INEQUALITY FOR MARTINGALES[J].Acta mathematica scientia,Series B, 2019, 39(2): 413-419.

Trendmd

References

[1] Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function. Trans Amer Math Soc, 1972, 165:207-226
[2] Kerman R, Torchinsky A. Integral inequalities with weights for the Hardy maximal function. Studia Math, 1981, 71:277-284
[3] Sawyer E T. A characterization of a two weight norm inequality for maximal operators. Studia Math, 1982, 75:1-11
[4] Bagby R. Weak bounds for the maximal function in weighted Orlicz spaces. Studia Math, 1990, 95:195-204
[5] Bloom S, Kerman R. Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal operator. Studia Math, 1994, 110(2):149-167
[6] Lai Q S. Two weight Φ-inequalities for the Hardy operator,Hardy-Littlewood maximal operator, and fractional integrals. Proc Amer Math Soc, 1993, 118(1):129-142
[7] Gogatishvili A, Kokilashvili V. Necessary and sufficient conditions for weighted Orlicz class inequalities for maximal functions and singular integrals. I. Georgian Math, 1995, 2(4):361-384
[8] Pick L. Two-weight weak type maximal inequalities in Orlicz classes. Studia Math, 1991, 100(3):207-218
[9] Kikuchi M. On weighted weak type maximal inequalities for martingales. Math Inequalities Appl, 2003, 6(1):163-175
[10] Ren Y B, Hou Y L. Two-weight weak-type maximal inequalities for martingales. Acta Math Sci, 2009, 29B(2):402-408
[11] Chen W, Liu P D. Several weak-type weighted inequalities in Orlicz martingale classes. Acta Math Sci, 2011, 31B(3):1041-1050
[12] Chen W, Liu P D. Weighted norm inequalities for multisublinear maximal operator in martingale spaces. Tohoku Math J, 2014, 66(4):539-553
[13] Chen W, Liu P D. Weighted integral inequalities in Orlicz martingale classes. Sci China Math, 2011, 54(6):1215-1224
[14] Osekowski A. Weighted maximal inequalities for martingales. Tohoku Math J, 2013, 65(1):75-91
[15] Osekowski A. Sharp L^p-bounds for the martingale maximal function. Tohoku Math J, 2018, 70(1):121-138
[16] Long R L. Martingale Spaces and Inequalities. Beijing:Peking Univ Press, 1993
[17] Weisz F. Martingale Hardy Spaces and their Applications in Fourier Analysis. New York:Springer-Verlag, 1994