【作者】 陈伟；

【导师】 刘培德；

【作者基本信息】 武汉大学 ， 基础数学， 2010， 博士

【摘要】 上世纪七十年代,随着欧氏空间中Ap权理论的建立,人们对加权理论有了新的认识.八十年代,在函数空间中有关Ap权性质的深刻结论迅速建立,比如Ap权的因子分解,向量值函数加权不等式,外插理论等等.随后,在Orlicz函数类,Lorentz空间,重排不变空间以及Musielak-Orlicz函数类上,加权不等式的研究相继展开,成果丰硕.在鞅空间中,加权不等式的研究也始于上世纪七十年代,但是进展一直十分缓慢.事实上,与欧式空间相比,概率空间既无代数结构也无拓扑结构,从而,依赖于欧式空间代数与拓扑结构的各种分解与覆盖定理不再适用.因此,在鞅的加权不等式理论研究中,必须使用新的工具和方法.本文围绕鞅空间中的加权不等式理论展开,研究某些算子的加权不等式及其成立的充分必要条件.我们考察的算子主要是极大算子,广义极大算子,几何极大算子.在建立各算子的加权不等式时,相应工作各有侧重.就极大算子而言,相关内容比较丰富.在Orlicz鞅类中,我们分别刻画了如下类型的加权不等式：当φ函数性质比较差时,我们建立了弱型和超弱型模不等式；当φ函数性质比较好时,我们建立了强型和弱型积分不等式.如果φ函数性质很差,那么弱型模不等式成立当且仅当所涉及的权是A1权,该结论与已有结论吻合——极大算子是由L1(u)到wL1(v)的有界算子当且仅当(u,u)∈A1.另外,针对一类特殊的双指标鞅空间,以外插为工具,我们建立了混合范数加权不等式.我们将广义极大算子引入鞅空间,给出了该算子加权不等式的刻画.特别地,当p＞1时,M是由Lp(Ω)到Lp(Ω×N,μ)或w Lp(Ω×N,μ)的有界算子当且仅当u是Ω×N上的Carleson测度.在鞅论中,关于Carleson测度的理论得到了进一步的丰富.与极大算子不同,几何极大算子既不是次线性算子,也不是拟线性算子.因此,该算子的内插理论缺失.在Orlicz鞅类中,借助几何极大算子的特性,我们刻画了一类加权积分不等式.当φ是幂函数时,我们的结论退化为(p,q)情形.此时,几何极大算子的特性导致不同指标的不等式之间的相通性.一方面,该算子的加权不等式本质上依赖于q/η而不是单纯的依赖指标p与q；另一方面,该算子是由L1到L1的有界算子.针对该算子,建立加权不等式的时候,我们也关注了A∞权的性质.在特定的前提下,我们给出了A∞权的各种等价刻画.本学位论文共有五章：第一章,回顾鞅空间理论发展的历程,介绍加权不等式理论的发展现状,阐述学位论文的选题意义以及创新点.第二章,基本概念和基本结论的汇总.第三章,在鞅空间中,给出了A∞权的各种等价定义.另外,在双指标鞅空间中,就一类特殊的双指标鞅空间建立了极大算子的加权不等式.第四章,在Orlicz鞅类中,讨论极大算子加权不等式成立的充分必要条件.此外,将广义极大算子引入鞅空间,建立了该算子与Carleson测度的联系.第五章,直接考察几何极大算子与权条件的关系,建立了几何极大算子的加权积分不等式.更多还原

【Abstract】 The theory of Ap was spurred in the 1970s, from which a better understand-ing of weighted inequalities was obtained. Subsequently, the relevant subjects were quickly established in Rn, for example, factorization of Ap weights, vector-valued inequalities, extrapolation of operators, and so on. Recently, weighted inequalities were widespreadly studied in Orlicz function classes, Lorentz spaces, rearrangement invariant spaces, as well as Musielak-Orlicz function classes. In martingale spaces, weighted inequalities first appeared in 1970s, but they has been developing slowly. One reason is that some decomposition theorems and covering theorems which depend on algebraic structure and topological structure are invalid on probability space.In this paper, weighted inequalities and characterizations of weights are obtained in martingale spaces. The maximal operator, the general maximal operator and the maximal geometric mean operator are extensively studied.For the maximal operator, we construct different types of weighted inequali-ties relying on the property ofΦfunction in Orlicz martingale classes. Sometimes, we are interested in some weak inequality as well as extra-weak modular inequal-ity and at other times we are interested in integral inequalities. Specially, in some extreme case, the weak modular inequality is valid if and only if the weights are A1 ones, which coincides the well known result that the maximal operator is bounded from L1(u) to wL1(v) if and only if (u,v)∈A1. Moreover, for a special class of two-parameter martingale spaces, exploiting extrapolation theory as a tool, we obtain a mixed-norm weighted inequality.We introduce the general maximal operator M into martingale space, and character the weighted inequality, too. In fact, if p>1, then M maps Lp(Ω) to Lp(Ω×N,μ) or wLp(Ω×N,μ) if and only ifμis a Carleson measure onΩ×N. In martingale setting, the theory of Carleson measure is further enriched by us.Compared with the maximal operator, the maximal geometric mean oper-ator is neither a sub-linear operator nor a quasi-linear operator. Therefore, the interpolation theory is invalid for it. Considering its feature, we obtain some weighted integral inequalities in Orlicz martingale classes. WhenΦis a power function, our inequalities reduce to the (p, q) ones, which are very interesting. In fact, the weighted inequalities depend only on the q/p rather than depend solely on indicators of p and q; on the other hand, the operator maps L1 to L1. While studying the maximal geometric mean operator, we also focus our attention on the A∞weight. On the assumption thatω∈S, we give a variety of equivalent definitions ofω∈A∞.The paper is divided into five chapters. The first one surveys results of mar-tingale spaces and weights. It also contains the significance of the dissertation. Chapter 2 consists of preliminaries. The following Chapter 3 discusses equiva-lent definitions of A∞weight. In addition, a class of two-parameter martingale spaces is studied. Chapter 4 is devoted to weighted inequalities for the maximal operator in Orlicz martingale classes. Moreover, the Carleson measure and the general maximal operator are also considered. The last chapter deals with the maximal geometric mean operator.更多还原