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Probab. Theory Relat. Fields (2012) 152:179206 DOI 10.1007/s00440-010-0319-2
Interpolation and [Phi1]-moment inequalities of noncommutative martingales
Turdebek N. Bekjan Zeqian Chen
Received: 4 February 2010 / Revised: 21 August 2010 / Published online: 18 September 2010 Springer-Verlag 2010
Abstract This paper is devoted to the study of -moment inequalities for noncom-mutative martingales. In particular, we prove the noncommutative -moment analogues of martingale transformations, Steins inequalities, Khintchines inequalities for Rademachers random variables, and BurkholderGundys inequalities. The key ingredient is a noncommutative version of Marcinkiewicz type interpolation theorem for Orlicz spaces which we establish in this paper.
Keywords -Measurable operators Noncommutative martingale Interpolation
-Moment martingale inequality Noncommutative Orlicz space
Mathematics Subject Classication (2000) 46L53 46L52 60G42
0 Introduction
Given a probability space ( , F , P). Let {Fn}n1 be a nondecreasing sequence
of -subelds of F such that F = Fn, and let be an Orlicz function with
T. Bekjan was partially supported by NSFC grant No. 10761009 and Z. Chen was partially supported by NSFC grant No. 10775175.
T. N. BekjanCollege of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China
Z. Chen (B)
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, West District 30, Xiao-Hong-Shan, Wuhan 430071, Chinae-mail: [email protected]
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180 T. N. Bekjan, Z. Chen
1 < p q < . If f = ( fn)n1 is a L -bounded martingale, then
n=1
|d fn|2
1
2
d P sup n1
(| fn|)d P, (0.1)
where d f = (d fn)n1 is the martingale difference of f and depends only on .
This result is the well-known BurkholderGundy inequality for convex powers (t) =
t p (see [11]) and proved in the general setting of Orlicz functions by Burkholder DavisGundy [10]. In their remarkable paper [34], Pisier and Xu proved the noncom-mutative analogue of the BurkholderGundy inequality, which triggered a systematic research of noncommutative martingale inequalities (we refer to a recent book by Xu [40] for an up-to-date exposition of theory of noncommutative martingales). In this paper, we will extend their work to -moment versions, i.e., we will prove the noncommutative analogue of (0.1).
Let us briey describe our -moment inequality. Let M be a nite von Neumann
algebra with a normalized normal faithful trace , and {Mn}n0 be an increasing ltra