涉及无穷乘积的加权不等式

发布时间：2018-04-14 05:12

本文选题：加权不等式 + 广义H(o|)lder不等式　；参考：《扬州大学》2016年硕士论文

【摘要】：上世纪七八十年代,欧氏空间中Ap权理论和Sp权理论建立之后,人们对加权理论保持着持续的关注和研究.在研究加权理论时,二进方法一直发挥着重要作用.从几何的观点上解释,我们可以对空间进行非常精细而且互相嵌套的划分.从概率论上解释,我们可以充分利用鞅空间中的各种不等式.近几年,加权理论的多线性问题的研究正蓬勃兴起.多线性问题的研究,很大程度上得益于二进方法.针对Ap权的多线性问题,建立二进系统,可以定量地研究极大算子对于Ap权常数的依赖.在某种意义下,甚至可以做到最优估计.但是,Sp条件衍生的多线性问题似乎比Ap条件衍生的问题复杂.尽管如此,假设某种单调性或逆向H6lder不等式,多线性版本的Sp加权理论仍然可以建立.受多线性加权理论的启发,本文将研究无穷线性(即涉及无穷乘积)加权不等式。我们定义了一个涉及无穷乘积的二进广义极大算子.针对该算子我们研究了涉及无穷乘积的加权不等式.具体地说,建立了相关的Carleson嵌入定理和弱型的广义Holder不等式,刻画了广义极大算子的弱型和强型加权不等式.本文的成果主要依赖于积分形式的广义Holder不等式和弱型的广义Holder不等式.我们的结果涉及无穷乘积,因此,必须注意无穷乘积的收敛问题.本文分为三章.第一章是引言,主要介绍了Rn中Hardy-Littlewood极大算子和多线性极大算子的加权不等式.第二章是预备知识,包含论文用到的一些基本定义和结果.特别地,我们给出了弱型的广义Holder不等式.第三章陈述并证明了我们的主要结果,即涉及无穷乘积的加权不等式.
[Abstract]:Since the establishment of Ap weight theory and Sp weight theory in Euclidean space in 1970s and 1980s, people have been paying more and more attention to weighting theory.The binary method has always played an important role in the study of weighting theory.From a geometric point of view, we can divide spaces into very fine and nested spaces.In terms of probability theory, we can make full use of all kinds of inequalities in martingale space.In recent years, the study of multilinear problems in weighted theory is booming.The study of multilinear problems is largely due to the binary method.To solve the multilinear problem of AP weight, a binary system is established to quantitatively study the dependence of the maximal operator on the AP weight constant.In a sense, even the best estimate can be achieved.But the multilinear problem of Sp conditional derivation seems to be more complicated than that of AP conditional derivation.Nevertheless, supposing some monotonicity or inverse H6lder inequality, the multilinear version of Sp-weighted theory can still be established.Inspired by the multilinear weighting theory, this paper will study the infinite linear (that is, infinite product) weighted inequalities.We define a dyadic generalized maximal operator involving infinite product.For this operator, we study the weighted inequalities involving infinite product.In particular, the Carleson embedding theorem and the generalized Holder inequality of weak type are established, and the weighted inequalities of weak type and strong type of generalized maximal operator are characterized.The results of this paper mainly depend on the integral form of generalized Holder inequality and weak type of generalized Holder inequality.Our results involve infinite product, so we must pay attention to the convergence of infinite product.This paper is divided into three chapters.The first chapter is the introduction, which mainly introduces the weighted inequalities of Hardy-Littlewood maximal operator and multilinear maximal operator in rn.The second chapter is the preparatory knowledge, including some basic definitions and results used in the paper.In particular, we give the generalized Holder inequality of weak type.In chapter 3, we present and prove our main results, that is, weighted inequalities involving infinite product.
【学位授予单位】：扬州大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O178

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