【作者】 李平；

【导师】 José Luis Torrea； 侯友良；

【作者基本信息】 武汉大学 ， 基础数学， 2017， 博士

【摘要】 在最近十年,半群理论被成功地应用到偏微分方程理论的研究中,最著名的成果之一是Caffarelli和Silvestre关于分数阶Laplace算子的工作.在最近几年,研究一些分数阶算子的性质是调和分析与偏微分方程理论的一个热门课题.本文的主要内容是利用抛物半群方法和抛物Calder6n-Zygmund理论研究抛物热方程和抛物调和震动发展方程,然后研究了联系这些抛物算子的半群的Littlewood-Paley g-函数理论和振动算子.本学位论文共有四章:第一章,我们回顾了分数阶算子、Littlewood-Paley理论和振动算子理论及其一些基本的抛物方程的发展历程及现状,叙述了本论文的选题的动机和意义,本文的研究方法和主要结果.第二章,首先建立了抛物半群理论.然后利用半群方法和抛物Calder6n-Zygmund理论研究热方程(?)tu-△u=f和抛物调和振动发展方程(?)tu-△u+|x|2=f,获得了方程解的加权混合范数Sobolev估计.我们也考虑了与这些抛物方程相应的Cauchy问题,证明了方程解的点态收敛性和加权混合范数Sobolev估计.第三章,利用抛物半群理论和向量值Calder6n-Zygmund理论来研究抛物算子(?)t-△和(?)t和△+|x|2的半群的Littlewood-Paley g-函数,证明了这些g-函数的Lp不等式.第四章,使用抛物半群语言和抛物Calderon-Zygmund理论研究抛物算子(?)t-△和(?)t-△+|x|2的Poisson半群的震动算子,证明了这些震动算子的Lp有界性,也考虑了震动算子在L∞空间中的局部增涨性.更多还原

【Abstract】 In the last decades the theory of semigroups have been used successfully in the development of a theory of PDEs,one of the most famous results is the fractional Laplacian operator which was studied by Caffarelli and Silvestre.And now,the theory related with fractional operators becomes a hot topic in Harmonic Analysis and PDEs.The main aim of this thesis is to development the parabolic semigroup theory to deal with heat equation and harmonic oscillator evolution equation.Then,we studied the Littlewood-Paley g-functions of semigroups associated with these parabolic oper-ators,and oscillation operators of Poisson semigroups associated with these parabolic operators also considered.This thesis is consist of four chapters.In Chapter 1,we recalled the development of fractional operators,Littlewood-Paley theory,oscillation operators and some basic parabolic equations,and presented the motivation of the thesis,and stated the research methods and main results of the thesis.In Chapter 2,we first developed the parabolic semigroups theory.Then,we used the parabolic semigroups and parabolic vector-valued Calderon-Zygmund theory to study the heat equation(?)tu-△u=fand harmonic oscillator evolution equation(?)tu-△u+|x|2=fwe got the weighted mixed-norm Sobolev estimates of solutions associated with these equations.We also considered the corresponding Cauchy problems,proved a.e.conver-gence and weighted mixed-norm estimates for solutions of these equations above.In Chapter 3,we used the parabolic semigroups and parabolic vector-valued Calderon-Zygmund theory to consider the Littlewood-Paley g-functions of semigroups associated with parabolic operators(?)t-△ and(?)t-△+|x|2,proved the Lp inequalities of these g-functions.In Chapter 4,considered oscillation operators of Poisson semigroups associated with parabolic operators(?)t-△ and(?)t-△+|x|2;we proved the Lp boundedness of these oscillation operators by using the parabolic semigroups theory and vector-valued Calderon-Zygmund theory,the case L∞ of oscillation operators also considered.更多还原