A FOUR-WEIGHT WEAK TYPE MAXIMAL INEQUALITY FOR MARTINGALES

Yanbo REN

doi:10.1007/s10473-019-0207-5

Acta mathematica scientia, Series B >

2019 , Vol. 39 >Issue 2: 413 - 419

DOI: https://doi.org/10.1007/s10473-019-0207-5

Articles

A FOUR-WEIGHT WEAK TYPE MAXIMAL INEQUALITY FOR MARTINGALES

Yanbo REN

Expand

School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, China

Received date: 2018-03-16

Revised date: 2018-06-11

Online published: 2019-05-06

Supported by

Supported by the National Natural Science Foundation of China (11871195).

Fold

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

Cite this article

Yanbo REN . A FOUR-WEIGHT WEAK TYPE MAXIMAL INEQUALITY FOR MARTINGALES[J]. Acta mathematica scientia, Series B, 2019 , 39(2) : 413 -419 . DOI: 10.1007/s10473-019-0207-5

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

Options

Abstract

Outlines

模态框（Modal）标题

Abstract

Cite this article

References