变指数空间中Hardy算子相关性质研究

发布时间：2017-12-27 05:04

本文关键词：变指数空间中Hardy算子相关性质研究　出处：《大连海事大学》2017年硕士论文　论文类型：学位论文

【摘要】：变指数函数空间由于其在解决自然科学、流体力学尤其是电变流体的研究、图像恢复以及工程技术中的一些非线性增长问题等方面发挥了有效的作用,近二十年来吸引了众多国内外学者的关注,得到了迅速的发展,已出现许多重要的成果。其中,Hardy算子作为函数空间中的经典算子,人们对于它的研究一直在持续之中。本学位论文只是针对Hardy算子的部分性质做了相应的工作,探究了 Hardy算子及加权Hardy算子在变指数空间中的有界性。本文组织框架如下:第一章是绪论,简单介绍了变指数空间的研究背景与研究意义,说明了本文所需要的一些基本知识和记号以及相关的引理,最后简要叙述了本学位论文的主要工作。第二章证明了 Hardy算子的对偶算子Cesaro算子在变指数Lebesgue空间中的有界性,证明思路与Hardy算子的证明类似,通过获得关于因子悲β(x)几乎增的估计,再利用Holder不等式等刻画工具得到Cesaro算子有界。第三章介绍了变指数Hardy算子,对保留xβ(·)形式的Hardy型不等式提出了一个充分必要条件,并证明了该Hardy算子在变指数Herz空间中的有界性。根据Herz空间的特征刻画,将变指数Herz空间转化为我们熟悉的变指数Lebesgue空间,再由Herz空间包含Lebesgue空间的关系获得x-1/p'(x)+β(x)+∈是几乎减的估计进一步得到Hardy算子有界。第四章进一步推广了的Hardy不等式的结果,将Hardy算子推广至加权Hardy算子,通过引入两个测试函数分别获得关于权函数ψ(x)和因子β(x)的估计,利用Holder不等式、Minkowskin不等式等刻画工具获得加权Hardy算子在变指数Lebesgue空间中的有界条件。
[Abstract]:Because of the space variable exponent function in solving natural science, fluid mechanics especially play an effective role on Electrorheological research, image restoration and engineering technology of some nonlinear growth problems, over the past twenty years has attracted many domestic and foreign scholars, has been rapid development, there have been many important results. Among them, the Hardy operator is a classical operator in the function space, and the research on it has been continuing. This thesis only works for some properties of Hardy operator, and explores the boundedness of Hardy operator and weighted Hardy operator in variable exponent space. The framework of this paper is as follows: the first chapter is the introduction, which briefly introduces the research background and research significance of variable exponent space, illustrates some basic knowledge and notations, and related lemmas. Finally, it briefly describes the main work of this dissertation. The second chapter proves that the dual operator Cesaro operator Hardy operator in Lebesgue space with variable index of boundedness, proof of concept with Hardy operator is similar to that obtained by a factor beta (x) estimates almost sad increase, then use Holder inequality characterization tool to get Cesaro is bounded. In the third chapter, we introduce the variable exponent Hardy operator, and give a necessary and sufficient condition for the Hardy type inequality preserving the X beta (?) form, and prove the boundedness of the Hardy operator in the variable exponent Herz space. According to the characteristics of characterizations of Herz spaces, the variable exponent Herz space into variable index we are familiar with the Lebesgue space, the relationship by Herz space contains Lebesgue space x-1/p'(x) + beta (x) +, is further reduced to estimate almost bounded operator Hardy. The fourth chapter further generalize the Hardy inequality of the results will be extended to the weighted Hardy operator Hardy operator, through the introduction of the two test functions were obtained on the right function (x) and factor beta (x) estimation, weighted Hardy operator in characterization tool variable exponent Lebesgue space by using the Holder inequality and Minkowskin inequality. Bounded condition.
【学位授予单位】：大连海事大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177

【相似文献】