本文選題：加權(quán)不等式 + 廣義H(o|)lder不等式　； 參考：《揚(yáng)州大學(xué)》2016年碩士論文

【摘要】：上世紀(jì)七八十年代,歐氏空間中Ap權(quán)理論和Sp權(quán)理論建立之后,人們對(duì)加權(quán)理論保持著持續(xù)的關(guān)注和研究.在研究加權(quán)理論時(shí),二進(jìn)方法一直發(fā)揮著重要作用.從幾何的觀點(diǎn)上解釋,我們可以對(duì)空間進(jìn)行非常精細(xì)而且互相嵌套的劃分.從概率論上解釋,我們可以充分利用鞅空間中的各種不等式.近幾年,加權(quán)理論的多線性問題的研究正蓬勃興起.多線性問題的研究,很大程度上得益于二進(jìn)方法.針對(duì)Ap權(quán)的多線性問題,建立二進(jìn)系統(tǒng),可以定量地研究極大算子對(duì)于Ap權(quán)常數(shù)的依賴.在某種意義下,甚至可以做到最優(yōu)估計(jì).但是,Sp條件衍生的多線性問題似乎比Ap條件衍生的問題復(fù)雜.盡管如此,假設(shè)某種單調(diào)性或逆向H6lder不等式,多線性版本的Sp加權(quán)理論仍然可以建立.受多線性加權(quán)理論的啟發(fā),本文將研究無窮線性(即涉及無窮乘積)加權(quán)不等式。我們定義了一個(gè)涉及無窮乘積的二進(jìn)廣義極大算子.針對(duì)該算子我們研究了涉及無窮乘積的加權(quán)不等式.具體地說,建立了相關(guān)的Carleson嵌入定理和弱型的廣義Holder不等式,刻畫了廣義極大算子的弱型和強(qiáng)型加權(quán)不等式.本文的成果主要依賴于積分形式的廣義Holder不等式和弱型的廣義Holder不等式.我們的結(jié)果涉及無窮乘積,因此,必須注意無窮乘積的收斂問題.本文分為三章.第一章是引言,主要介紹了Rn中Hardy-Littlewood極大算子和多線性極大算子的加權(quán)不等式.第二章是預(yù)備知識(shí),包含論文用到的一些基本定義和結(jié)果.特別地,我們給出了弱型的廣義Holder不等式.第三章陳述并證明了我們的主要結(jié)果,即涉及無窮乘積的加權(quán)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.
【學(xué)位授予單位】：揚(yáng)州大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類號(hào)】：O178

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