ATOMIC DECOATOMIC DECOMPOSITIONS AND DUALS OF WEAK HARDY SPACES OF B-VALUED MARTINGALES

In this article, several weak Hardy spaces of Banach-space-valued martingales are introduced, some atomic decomposition theorems for them are established and their duals are investigated. The results closely depend on the geometrical properties of the Banach space in which the martingales take values.

CONVERGENCE OF WEIGHTED AVERAGES OF MARTINGALES IN NONCOMMUTATIVE BANACH FUNCTION SPACES

Let x （xn）≥1 be a martingale on a noncommutative probability space n （M, r） and （wn）n≥1 a sequence of positive numbers such that Wn = ∑ k=1^n wk →∞ as n →∞ We prove that x = （x.）n≥1 converges in E（M） if and only if （σn（x）n≥1 converges in E（.hd）, where E（A//） is a noncommutative rearrangement invariant Banach function space with the Fatou property and σn（x） is given by σn（x） = 1/Wn ∑k=1^n wkxk, n=1, 2, .If in addition, E（Ad） has absolutely continuous norm, then, （an（x））≥1 converges in E（.M） if and only if x = （Xn）n≥1 is uniformly integrable and its limit in measure topology x∞∈ E（M）.

Atomic Decompositions of Weak Hardy Spaces of B-Valued Martingales

The atomic decompositions of weak Hardy spaces of Banach-space-valued martingales are given. With the help of the atomic decompositions, some inequalities for B-valued martingales are established in the case 0〈r≤1. Here the results are connected closely with the p-uniform smoothness and q-uniform convexity of Banach spaces which the martingales take values in.

CARLESON MEASURES AND THE GENERALIZED CAMPANATO SPACES OF VECTOR-VALUED MARTINGALES

In this paper, the so-called（p,Ф）-Carleson measure is introduced and the rela-tionship between vector-valued martingales in the general Campanato spaces Lp,Ф（X） and the （p, Ф）-Carleson measures is investigated. Specifically, it is proved that for q ∈ [2, ∞）, the measure d# ：-=││ dfk││^qdP dm is a （q, Ф）-Carleson measure on Ω × N for every f ∈ Lq,Ф（X） if and only if X has an equivalent norm which is q-uniformly convex; while for p C （1, 2], the measure dμ ：=││dfk││^pP dm is a （p, Ф）-Carleson measure on Ω ×N implies that f ∈ Lp,Ф（X） if and only if X admits an equivalent norm which is p-uniformly smooth. This result extends an earlier result in the literature from BMO spaces to general Campanato spaces.

NEW DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES

Let 1≤q≤∞,b be a slowly varying function and letΦ:[0,∞)■[0,∞)be an increasing convex function withΦ(0)=0 and■Φ(r)=∞.In this paper,we present a new class of Doob’s maximal inequality on Orlicz-Lorentz-Karamata spaces LΦ,q,b.The results are new,even for the Lorentz-Karamata spaces withΦ(t)=tp,the Orlicz-Lorentz spaces with b≡1,and weak Orlicz-Karamata spaces with q=∞in the framework of LΦ,q,b-Moreover,we obtain some even stronger qualitative results that can remove the△2-condition of Liu,Hou and Wang(Sci China Math,2010,53(4):905-916).

CONDITIONAL EXPECTATIONS AND MARTINGALES IN NONCOMMUTATIVE Lp-SPACES ASSOCIATED WITH CENTER-VALUED TRACES

In this paper we prove the existence of conditional expectations in the noncom- mutative Lp（M, Ф）-spaces associated with center-valued traces. Moreover, their description is also provided. As an application of the obtained results, we establish the norm convergence of weighted averages of martingales in noncommutative Lp（M, Ф）-spaces.

Hàjek-Rèniy Type Inequality for Banach Space Valued Martingales and Its Applications

In this paper,we establish the Hàjek-Rèniy type inequality for Banach space valued martingales generalizing the recent results of Tómcs and L'ibor [1].Then p-uniformly smoothable Banach space is characterized in terms of the Hàjek-Rèniy type inequality for Banach space valued martingales.Those results generalize the recent results of Gan Shixin [2].

BANACH SPACE VALUED QUASI-EVENTUAL MARTINGALES

This article investigates convergence, transforms and q-square summability of banach space valued quasi-eventual martingales. Some basic results of Banach space valued martingales are improved and extended.

DOOB’S MAXIMAL INEQUALITIES FOR MARTINGALES IN VARIABLE LEBESGUE SPACE

In this paper we deal with the martingales in variable Lebesgue space over a probability space.We first prove several basic inequalities for conditional expectation operators and give several norm convergence conditions for martingales in variable Lebesgue space.The main aim of this paper is to investigate the boundedness of weak-type and strong-type Doob’s maximal operators in martingale Lebesgue space with a variable exponent.In particular,we present two kinds of weak-type Doob’s maximal inequalities and some necessary and sufficient conditions for strong-type Doob’s maximal inequalities.Finally,we provide two counterexamples to show that the strong-type inequality does not hold in general variable Lebesgue spaces with p>1.

LORENTZ MARTINGALE SPACES AND INTERPOLATION

In this article, the authors introduce some new Lorentz spaces for martingales, which are extensions of Hardy spaces of martingales. Then they discuss their basic properties, embedding relationships, and interpolation spaces between them, during which the use of rearrangement good-λ-inequality plays an important role.

BURKHOLDER-GUNDY-DAVIS INEQUALITY IN MARTINGALE HARDY SPACES WITH VARIABLE EXPONENT

In this article, by extending classical Dellacherie＇s theorem on stochastic se- quences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis in- equality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier＇s inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.

INTERPOLATION OF LORENTZ-ORLICZ MARTINGALE SPACES

In this paper, we apply function parameters to real interpolation of Lorentz- Orlicz martingale spaces. Some new interpolation theorems are formulated which generalize some known results in Lorentz spaces An introduced by Sharpley.

WEAK ATOMIC DECOMPOSITIONS OF MARTINGALE HARDY-LORENTZ SPACES AND APPLICATIONS

In this article, we establish some atomic decomposition theorems for martin- gale Hardy-Lorentz spaces. As applications, with the help of weak atomic decompositions, some interpolation theorems for martingale Hardy-Lorentz spaces are proved.

BOUNDEDNESS OF DYADIC DERIVATIVE AND CESARO MEAN OPERATOR ON SOME B-VALUED MARTINGALE SPACES

In this article, it is proved that the maximal operator of one-dimensional dyadic derivative of dyadic integral I＊ and Cesàro mean operator σ＊ are bounded from the B-valued martingale Hardy spaces pΣα, Dα, pLα, p H#α, pKr to Lα （0 α ∞）, respectively. The facts show that it depends on the geometrical properties of the Banach space.

ON THE CONCAVE φ-INEQUALITIES FOR NONNEGATIVE SUBMARTINGALES

Let 1,2 be nonnegative nondecreasing functions, and 1 be concave. Theauthors prove the equivalence of the following two conditions:(i) E1(Mf) < cE2(Zo+A) for every nonnegative submartingale f = (fn)n>o with it'sDoob's Decomposition: f= Z + A, where Z is a martingale in L1 and A is a nonnegativeincrasing and predictable process.(ii) There exists positive constants c, to such that > to.When 1 =2 the condition (ii) above is equivalent to the classical condition p < 1. Asa consequence, for a concave function ,if and only if E1(Mf) < cE2(Zo+A)for every nonnegative submartingale f.

NONCOMMUTATIVE ORLICZ-HARDY SPACES

Let （Φ,Ψ） be a pair of complementary N-functions and HΦ（A） and HΨ（A） be the associated noncommutative Orlicz-Hardy spaces. We extend the Riesz, Szeg＆#168;o and inner-outer type factorization theorems of Hp（A） to this case.

DOOB'S INEQUALITY, BURKHOLDER-GUNDY INEQUALITY AND MARTINGALE TRANSFORMS ON MARTINGALE MORREY SPACES

We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob＇s inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob＇s inequality on martingale block spaces, we obtain the Davis＇ decompositions for martingale Morrey spaces.

Real Interpolation Between B-valued Martingale Hardy Spaces

The interpolation spaces between Banach space valued martingale Hardy spaces, between Hardy and BMO spaces are identified respectively. Some results obtained here are connected closely with the convexity and smoothness of the Banach space which the martingales take values in.

THE DYADIC DERIVATIVE AND CESRO MEAN OF BANACH-VALUED MARTINGALES

In this article, the Banach space X and the martingales with values in it are considered. It is shown that the maximal operators of the one-dimensional dyadic derivative of the dyadic integral and Cesaro means are bounded from the dyadic Hardy- Lorentz space pH^-ra（X） to Lra（X） when X is isomorphic to a p-uniformly smooth space （1 〈p ≤ 2）. And it is also bounded from Hra（X） to Lra（X） （0 〈 r 〈 ∞,0 〈 a≤oc） when X has Radon-Nikodym property. In addition, some weak-type inequalities are given.

Some Clifford Martingale Spaces

We study the spacesH p, Sp, Kp, Σ p and