Acta Mathematica Scientia

, Volume 39, Issue 2, pp 413–419 | Cite as

A Four-Weight Weak Type Maximal Inequality for Martingales

  • Yanbo RenEmail author


In this article, some necessary and sufficient conditions are shown in order that weighted inequality of the form
$$\int_{\left\{ {f* > \lambda } \right\}} {\Phi _1 \left( {\lambda w_1 } \right)w_2 d\mathbb{P}} \leqslant C\int_\Omega {\Phi _2 \left( {C\left| {f_\infty } \right|w_3 } \right)w_4 d} \mathbb{P}$$
holds a.e. for uniformly integrable martingales f = (fn)n≥0 with some constant C > 0, where Φ1, Φ2 are Young functions, wi (i = 1, 2, 3, 4) are weights, \(f* = \mathop {\sup }\limits_{n \geqslant 0} \left| {f_n } \right|\) and \(f_\infty = \mathop {\lim }\limits_{n \to \infty } f_n\) 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 

2010 MR Subject Classification

60G42 60G46 


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Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHenan University of Science and TechnologyLuoyangChina

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