WUJNS
Vol. 11 No. 3 2006 461-464
Wuhan University Journal of Natural Sciences
Article ID=1007 1202(2006)03-0461-04
A Biplurisubharmonic Characterization of AUMD Spaces 0
Introduction
[ ] ZHAO Wei, LIU Peide t School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China
Abstract=
We establish a new characterization of AUMI) (analytic unconditional martingale differences) spaces via hi plurisubharmonic functions. That is, B~- AUMD iff there ex isis a bpsbh (biplurisubharmonic) function L: B)< b--,'[--~,,="" ,~,)="" satisfying="" i.(.r,0),="" i="" .="" (="" 0="" ,="" y="" )="" ~="" i="" .="" (="" 0="" ,="" 0="" )="" ~="" 0="" ,="" i.(.r,y)~i.="" (0,0)§="" and="" l="" (="" .="" r="" ,="" y="" )="" ~="" ]="" .="" r="" .y]="" for="" i,+yl+l.~-="" yl="" ~="" 1="" .="" this="" provides="" an="" analogue="" of="" piasecki's="" characterization="" of="" aumi)="" spaces.="" ()ur="" arguments="" are="" based="" on="" some="" special="" properties="" of="" zigzag="" analytic="" martingales="" and="" martingale="">
Key words: AUMD space; analytic martingale; biplurisubharmonic function; martingale transform ct.c number= () 211.6
T
The basic properties of the fundamental class of Banach spaces having the UMD (unconditional martingale differences) property were established by BurkholderEl'e? Later, GarlingF:~I introduced the AUMD property for complex Banach spaces. Piasecki I*? obtained some geometric conditions for the AUMD spaces by using skew-plurisubharmonic functions recently. In this paper we use hiplurisubharmonic functions to characterize the AUMD property of a complex Banach space. The main method combined what was used in Ref. [1 and Ref. E4~.
1
Preliminaries
Let ~2= EO, 2rr~'~ and P be the product measure of normalized I.ebesgue measures on [-0,2~. An element 0~ a2 is written as 0= (0t ,0e ,'"). Let Z,, stands for the o-algebra generated by the first n coordinates 01,02 ~,*'",~5~n, and Zo = {~,E0, 2rr~}. I.et E be the expectation with respect to P. Suppose that B is a complex Banach space with its norm denoted by ] 9 I. A B-valued sequence F = (F,,) of random variables adapted to the sub-o-algebra (2,) is called an analytic martingale if F0 = z,F,, = .r§ 2~t.(01 ~''~
1)e i0* ,for n>~ 1
k=l Received date= 2005-06-20 Foundation item: Supported by the National Natural Science Foun darien of China (10371093) Biography: ZHA() Wei (1977 ), male, Ph. l). candidate, research direction: martingale theory, geometry of Banaeh spaces, functional analysis, harmonic analysis. E mail.,
[email protected], corn t To whom correspondence should be addressed. E mail: pdliu(~whu. edu. cn
where .r,/?l ~ B, and/?k is a measurable Bochner integrable function of 01 ,Oz,'",& ~ which take values in B, k-- 1, 2, 9". A martingale G--(C-,, ) is said to be the martingale _+ 1transform of analytic martingale F--(F,,) by the sequence v-(v,,), where v,, = 1 or - 1 for each n, if 461
,2_," Go = x,G. = x + ~ v~dG ,for n ~ 1 k
1
here dFe =Fe-Fe_~ for k = l , 2, "... Recall that a complex Banach space B is said to have the analytic UMD property (B~ AUMD, in short) if there is a constant C~ > 0 for some p ( O ( p % ~ ) or any such p such that
ZIG. I"
Gzl f,,
1 (1) for every B-valued analytic martingale F = (F,,) and its • 1-transform G-- (G.) as above, where C~ is a constant depends only on p. Several equivalent conditions for the AUMD property are found in Ref. [3 6]. We use the customary notation F/ = suplF~ I ,F ~ = suplF,,land IIFII, = sup [[f,, [], k~n
n~-O
n~jO
for the B-valued martingale F = (F,). Let B be a complex Banach space. A f