【作者】 马媛；

【导师】 刘风；

【作者基本信息】 山东科技大学 ， 数学， 2025， 硕士

【摘要】 目前关于高阶极大交换子及其分数次情形的正则性尚未有研究。本文系统化地研究了高阶极大交换子在Triebel-Lizorkin空间、Besov空间和Sobolev空间上的有界性与连续性问题。本文主要基于上述三类函数空间的特征刻画,结合一阶差分方法进行研究。全文共分五章内容。第一章主要介绍本文的研究背景、意义、国内外研究现状、研究动机及创新点。第二章在Lipschitz符号函数条件下建立高阶极大交换子和分数次极大交换子在端点Sobolev空间上的弱梯度有界性与连续性。第三章主要建立三个结果:在一阶Sobolev符号函数条件下建立高阶极大交换子和分数次极大交换子在一阶Sobolev空间上的有界性;在Triebel-Lizorkin符号函数条件下建立高阶极大交换子在Triebel-Lizorkin空间上的有界性和连续性;在Besov符号函数条件下建立高阶极大交换子在Besov空间上的有界性和连续性。第四章在Lipschitz符号函数条件下建立高阶极大交换子在Triebel-Lizorkin空间、Besov空间与Sobolev空间上的有界性和连续性。第五章对此论文的研究工作做出了总结,并进一步介绍了今后拟研究的问题。更多还原

【Abstract】 At present,the regularity of higher order maximal commutators and their fractional cases has not been studied.In this thesis,we systematically study the boundedness and continuity of higher order maximal commutators on Triebel-Lizorkin,Besov and Sobolev Spaces.This thesis mainly conducts research based on the characteristic characterization of the above three types of function Spaces,combined with the first-order difference method.The full text is divided into five chapters.In chapter 1,we mainly introduce the research background,significance,domestic and foreign research status,research motivation and innovation.In chapter 2,the weak gradient boundedness and continuity of higher order maximal commutator and fractional maximal commutator on Sobolev space are established under Lipschitz symbolic function.In chapter 3,three results are established:the boundedness of higher order maximal commutators and fractional maximal commutators on first-order Sobolev Spaces is established under the condition of first-order Sobolev symbolic functions;the boundedness and continuity of higher order maximal commutator on Triebel-Lizorkin space are established under the condition of Triebel-Lizorkin symbolic function;the boundedness and continuity of higher order maximal commutators on Besov Spaces are established under Besov symbolic functions.In chapter 4,we establish the boundedness and continuity of higher order maximal exchangers on Triebel-Lizorkin space,Besov space and Sobolev space under the condition of Lipschitz symbolic function.In chapter 5,we summarize the research work of this thesis and further introduce the problems to be studied in the future.更多还原