凹函数定义的弱Orlicz-Hardy空间之间的鞅变换

本文研究了两个弱Orlicz-Hardy鞅空间中元素之间相互转换关系的问题.利用鞅变换的方法,证明了：设φ₁是凹Young函数,φ₂是凹或者凸Young函数,且q_φ1 > 0,0 < q_φ2≤p_φ2 <+∞,则当φ₁≤φ₂时,wH_φ1中的元素是wH_φ2中元素的鞅变换的结果,所得结果将已有的相关结论由强型空间（赋范空间）推广到弱型空间（赋拟范空间）.     

Orlicz-Hardy空间与BMO空间之间的鞅变换

以鞅变换为工具,刻画了Orlicz-Hardy鞅空间与BMO空间之间的相互关系.证明了如下结论:对任意上指标有限(等价于满足△_2-条件)的Young函数Φ,鞅f∈H_Φ{P_Φ,Q_Φ}的充分必要条件是,f是BMO∈{BMO_1,BMO_2}中某个鞅g的鞅变换.     

LΦ可料控制鞅Hardy-Orlicz空间之间的鞅变换

以鞅变换为工具,刻画了LΦ可料控制鞅的Hardy-Orlicz空间之间的相互关系,设Φ1和Φ2是两个Young函数,并在某种意义上Φ2强于Φ1(具体定义见正文),以构造性的方法证明了Hardy-Orlicz空间(D)Φ1中的鞅恰好是Hardy-Orlicz空间(D)Φ2中的鞅的鞅变换.所得的结果推广了已有文献中的相关结论.     

L~Φ可料控制鞅Hardy-Orlicz空间之间的鞅变换

以鞅变换为工具,刻画了L^Φ可料控制鞅的Hardy-Orlicz空间之间的相互关系,设Φ₁和Φ₂是两个Young函数,并在某种意义上Φ₂强于Φ1（具体定义见正文）,以构造性的方法证明了Hardy-Orlicz空间D_Φ1中的鞅恰好是Hardy-Orlicz空间D_Φ2中的鞅的鞅变换.所得的结果推广了已有文献中的相关结论.     

拟鞅变换与UMD空间

本文定义了拟鞅变换,将Burkholder关于变换的不等式推广到拟鞅,同时应用拟鞅变换的若干性质刻划了UMD空间的特征.     

鞅变换Ф-不等式

本文对于可预报鞅证明了鞅变换的均方函数、极大函数的Ф-不等式．并讨论了鞅变换的极大函数的^aLФ范数与鞅的aKФ范数的比较关系．     

弱Hardy鞅空间与鞅的弱原子分解

定义了一些弱Hardy鞅空间和3种类型的弱原子. 它们与经典的H_p鞅论中的Hardy鞅空间和原子形成对应. 然后证明了弱Hardy鞅空间上的3个弱原子分解定理. 利用鞅的弱原子分解, 给出了弱Hardy鞅空间上的次线性算子有界的一个充分条件. 利用这个条件, 得到关于鞅的一系列弱H_p范数不等式和弱(p,p)型不等式, 以及各个弱Hardy鞅空间的连续嵌入关系. 这些不等式是经典的H_p鞅论中基本不等式的弱型对应.     

鞅变换φ-不等式

本文对于可预报鞅证明了鞅变换的均方函数、极大函数的Φ 不等式 ,并讨论了鞅变换的极大函数的αLΦ 范数与鞅的αKΦ 范数的比较关系     

p-拟鞅变换

本文研究了p-拟鞅变换的问题.利用Banach空间的几何性质,得到一维和二维参数下拟鞅变换理论,推广了鞅变换的一些结果.     

B-值鞅Hardy空间与BMO空间的实内插

研究了B值鞅Hardy空间与BMO空间的实内插，并利用所得结果，讨论了鞅Hardy空间的内插空间的共轭问题。     

鞅Hardy空间与Hardy-Orlicz空间的鞅变换

利用鞅变换,刻画了鞅Hardy空间与Hardy-Orlicz空间之间的相互关系:当p_Φ+∞时,证明了Hardy-Orlicz空间H_Φ~s中的鞅是Hardy空间H_1~s中的鞅变换;反之,H_1~s中的鞅也是H_Φ~s中的鞅变换.所得结果推广了已有文献中的相应结论.     

Atomic Decompositions for Two-Parameter Vector-Valued Martingales and Two-parameter Vector-valued Martingale Spaces

In this paper atomic decompositions for two-parameter vector-valued martingales are given. With the help of the atomic decompositions the relations between the mutual embedding of two-parameter vector-valued martingale spaces and geometric properties of Banach spaces are investigated. Our study shows that geometric properties of Banach spaces determine the embedding of martingale spaces and conversely the latter can characterize the former. This revised version was published online in June 2006 with corrections to the Cover Date.     

Two-Dimensional Conjugate Martingale Transforms

Characterizations of the two-dimensional H ₁, BMO and VMO martingale spaces generated by bounded Vilenkin systems via conjugate martingale transforms are studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Logarithmic Growth for Matrix Martingale Transforms

An example is given of an operator weight W that satisfies thedyadic operator Hunt–Muckenhoupt–Wheeden condition for which there exists a dyadic martingale transform on L² (W) that is unbounded. The constructionrelates weighted boundedness to the boundedness of dyadic vectorHankel operators.     

Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

Let L be a linear operator in L ²(? ⁿ ) and generate an analytic semigroup {e ^?tL } _t??0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by ??(L)??(0,??]. Let ?? on (0,??) be of upper type 1 and of critical lower type $\widetilde{p}_{0}(\omega)\in(n/(n+\theta(L)),1]$ and ??(t)=t ^?1/?? ^?1(t ^?1) for t??(0,??). In this paper, the authors first introduce the VMO-type space VMO _??,L (? ⁿ ) and the tent space $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ and characterize the space VMO _??,L (? ⁿ ) via the space $T^{\infty}_{\omega,\mathrm{v}}({{\mathbb{R}}}^{n+1}_{+})$ . Let $\widetilde{T}_{\omega}({{\mathbb{R}}}^{n+1}_{+})$ be the Banach completion of the tent space $T_{\omega}({\mathbb{R}}^{n+1}_{+})$ . The authors then prove that $\widetilde{T}_{\omega}({\mathbb{R}}^{n+1}_{+})$ is the dual space of $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ . As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{\rho,L^{\ast}}({\mathbb{R}}^{n})$ is the space B _??,L (? ⁿ ), where L ^* denotes the adjoint operator of L in L ²(? ⁿ ) and B _??,L (? ⁿ ) the Banach completion of the Orlicz-Hardy space H _??,L (? ⁿ ). These results generalize the known recent results by particularly taking ??(t)=t for t??(0,??).     

Martingale Transforms and the Hardy-Littlewood-Sobolev Inequality for Semigroups

We give a representation of the fractional integral for symmetric Markovian semigroups as the projection of martingale transforms and prove the Hardy-Littlewood-Sobolev (HLS) inequality based on this representation. The proof rests on a new inequality for the fractional Littlewood-Paley g–function.     

Martingale Transforms and Their Projection Operators on Manifolds

We prove the boundedness on L ^p , 1?<?p?<?∞, of operators on manifolds which arise by taking conditional expectation of transformations of stochastic integrals. These operators include various classical operators such as second order Riesz transforms and operators of Laplace transform-type.     

Brownian Representations of Cylindrical Local Martingales,Martingale Problem and Strong Markov Property of Weak Solutions of SPDEs in Banach Spaces

The paper deals with three issues. First we show a sufficient condition for a cylindrical local martingale to be a stochastic integral with respect to a cylindrical Wiener process. Secondly, we state an infinite dimensional version of the martingale problem of Stroock and Varadhan, and finally we apply the results to show that a weak existence plus uniqueness in law for deterministic initial conditions for an abstract stochastic evolution equation in a Banach space implies the strong Markov property.