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.

Some Clifford Martingale Spaces

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

Interpolation theorems on weighted Lorentz martingale spaces

In this paper several interpolation theorems on martingale Lorentz spaces are given.The proofs are based on the atomic decompositions of martingale Hardy spaces over weighted measure spaces.Applying the interpolation theorems,we obtain some inequalities on martingale transform operator.

B-valued martingale spaces with small index and atomic decompositions

Some atomic decomposition theorems for Banach-space-valued martingales are established. Using them, the embedding relationships between martingale spaces with small index are discussed. The results obtained here are connected closely with the p-uniform smoothness and q-uniform convexity of Banach space in which the martingales take values.

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.

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.

SEVERAL WEAK-TYPE WEIGHTED INEQUALITIES IN ORLICZ MARTINGALE CLASSES

The aim of this paper is to establish several necessary and sufficient conditions in order that the weighted inequality ρ（M f 〉 λ）Φ（λ） ≤ C ∫_Ω~Ψ （C｜f｜）σdμ,λ 〉 0 or ρ（Mf〉λ） ≤ C∫-Ω~Φ（Cλ^-1 ｜f｜）σdμ,λ 〉0 holds for every uniformly integral martingale f=（f_n）, where M is the Doob＇s maximal operator, Φ, Ψ are both Φ-functions, and e, σ are weights.

The Reverse Holder Classes in Martingale Setting

In martingale setting, it has been shown thatA p weights can be factorized in terms ofA 1 weights. This factorization benefits many problems very much. In this paper, the new class of RH∞ plays the same role for RH s , which makes the reverse H?lder inequalities hold with exponents>1, that the classA 1 does forA p class. Therefore, the Jones' factorization theorem forA p weights was extended to include some information about the reverse H?lder classes. And it is the most convenient object in weight theory indeed. Key words martingale space - minimal function - weight inequality - reverse H?lder class CLC number O 221. 4 Foundation item: Supported by the National Natural Science Foundation of China (19771063)Biography: Zuo Hong-liang (1976-), male, Ph. D candidate, research direction: martingale theory.

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.

The Hjek-Rnyi Inequlity For Banach Space Valued Random Variable Sequences and Its Application

In this paper we prove the following Hajek Renyi inequality:Let 0<p≤1 ,then for any Banach space B , any L p integrable B valued random variable sequence {D n,n≥1} ,any real number sequence {b n,n≥1} with 0<b n↑∞ ,any integer n≥1 ,there exits a constant C=C p>0 (only depending on p ) such thatP( sup j≥nji=1D ib j≥ε)≤Cε -p (∞j=n+1E‖D j‖ pb p j+nj=1E‖D j‖ pb p n) In the other direction,we prove some strong laws of large numbers and the integrability of the maximal functions for B valued random variable sequences by using this inequality and the Hajeck Renyi inequality we have obtained recently.Some known results are extended and improved.

APPROXIMATION BY WALSH-KACZMARZ-FEJR MEANS ON THE HARDY SPACE

The main aim of this paper is to find necessary and sufficient conditions for the convergence of Walsh-Kaczmarz-Fej′er means in the terms of the modulus of continuity on the Hardy spaces Hp, when 0〈p≤1/2.

WEIGHTED AVERAGE OF DIRICHLET KERNELS FOR VILENKIN-LIKE SYSTEM

For Vilenkin-like system, the authors define a new operator H*f := supn |Hnf|, where Hnf is the weighted average for partial sums, and prove that H* is of type (Hp* (Gm), Lp(Gm)) for all 1/2 < p ≤ ∞. As a consequence, the authors prove the operator S*f := supn |Snf| is of type (p, p) for 1 < p < ∞, where Snf is the n-partial sum.

B-Valued Dyadic Derivative

The principles of the new maximal operator H＊ we defined are discussed. We prove that it is bounded from martingale Hardy-Lorentz L^Xp.q[0,1） to the Lorentz L^Xp.q[0,1） for 1/2〈 p〈∞, 0〈~ q ≤ ∞, where X is any Banach space. When the Banach space X has the RN property, the sequence dnHnf converges to f a.e. Meanwhile the convergence in L^Xp norm for 1≤p〈∞ is a consequence of that the family functions K （n∈N） is an approximate identity.

Some Strong Laws of Large Numbers for Blockwise Martingale Difference Sequences in Martingale Type p Banach Spaces

For a blockwise martingale difference sequence of random elements {Vn, n ≥ 1} taking values in a real separable martingale type p （1 ≤ p ≤ 2） Banach space, conditions are provided for strong laws of large numbers of the form limn→∞ Vi/gn = 0 almost surely to hold where the constants gn ↑∞. A result of Hall and Heyde [Martingale Limit Theory and Its Application, Academic Press, New York, 1980, p. 36] which was obtained for sequences of random variables is extended to a martingale type p （1〈 p ≤2） Banach space setting and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided.

Real Interpolation between Martingale Hardy and BMO Spaces

In this paper, we consider the real interpolation with a function parameter between martingale Hardy and BMO spaces. An interpolation theorem for martingale Hardy and BMO spaces is formulated. As an application, real interpolation between martingale Lorentz and BMO spaces is given. Keywords Martingale space, BMO space, Lorentz space, real interpolation, function parameter

ON ALMOST SURE CONVERGENCE OF WEIGHTED SUMS OF RANDOM ELEMENT SEQUENCES

We mainly study the almost sure limiting behavior of weighted sums of the form ∑ni=1 aiXi/bn , where {Xn, n ≥ 1} is an arbitrary Banach space valued random element sequence or Banach space valued martingale difference sequence and {an, n ≥ 1} and {bn,n ≥ 1} are two sequences of positive constants. Some new strong laws of large numbers for such weighted sums are proved under mild conditions.

A. S. Convergence of Two-Parameter Banach Space Valued Martingales and the Radon-Nikodym Property of Banach Spaces

In this paper, we prove that under the F4 condition, any L log+ L bounded two-parameter Banach space valued martingale converges almost surely to an integrable Banach space valued random variable if and only if the Banach space has the Radon-Nikodym property. We further prove that the above conclusion remains true if the F4 condition is replaced by the weaker local F4 condition.

REPRESENTATION OF MARTINGALE ADDITIVE FUNCTIONALS 0N BANACH SPACES

Assume that B is a separable real Banach space and X(t) is a diffusion process on B. In thispaper, we will establish the representation theorem of martingale additive functionals of X(t).

The Martingale Hardy Type Inequalities for Dyadic Derivative and Integral

Since the Leibniz-Newton formula for derivatives cannot be used in local fields, it is important to investigate the new concept of derivatives in Walsh-analysis, or harmonic analysis on local fields. On the basis of idea of derivatives introduced by Butzer, Schipp and Wade, Weisz has proved that the maximal operators of the one-dimensional dyadic derivative and integral are bounded from the dyadic Hardy space Hp,q to Lp,q, of weak type （L1,L1）, and the corresponding maximal operators of the two-dimensional case are of weak type （Hi, L1）. In this paper, we show that these maximal operators are bounded both on the dyadic Hardy spaces Hp and the hybrid Hardy spaces H^#p 0〈p≤1.

The Martingale Hardy Type Inequality for Marcinkiewicz-Fejer Means of Two-dimensional Conjugate Walsh-Fourier Series

The main aim of this paper is to prove that for any 0 〈 p≤ 2/3 there exists a martingale f E Hp such that Marcinkiewicz Fejer means of the two-dimensional conjugate Walsh Fourier series of the martingale f is not uniformly bounded in the space Lp.