The Weighted Estimation Of The Commutators Of Unilateral Extremely Large Operators And A Class Of Bilinear Square Operators

In this paper,we mainly introduce the weighted boundedness of two kinds of operator commutators.Firstly,we prove one-sided weighted strong type inequalities for commutators generated by one-sided operators and Lipschitz functions.Moreover,a sufficient and neces-sary condition for the corresponding boundedness of commutators of two kinds of one-sided maximal operators on the Triebel-Lizorkin spaces will be given.Secondly,under weaker kernel assumption,the author discusses the boundedness of bilinear square functions asso-ciated with non-smooth kernels on the product of weighted Lebesgue spaces.Moreover,we consider the weighted boundedness of the commutators of bilinear square functions on the product of weighted Lebesgue spaces,with symbols in BMO and weighted BMO respectively.Considering the generality of the kernel,as an application,some known results are extended.It consists of four chapters:In chapter 1,we introduce the research background of weighted boundedness properties of Commutators of two types of operators and their related basic definitions:the basic definitions of Triebel-Lizorkin spaces,unilateral operators such as Hardy-Littlewood maximal operators in other definitions,basic definitions of non-convolutional bilinear Marcinkiewicz integrals and bilinear Littlewood-Paley g-functions,and definitions of weight functions and unilateral weights.One-sided weighted Triebel-Lizorkin space,symbolic function belongs to Lipschitz space,weighted BMO space and Morrey space.In chapter 2,we first introduce several auxiliary lemmas and then give the prove of Lipschitz estimates for one-sided higher order commutators on weighted boundedness one-sided Triebel-Lizorkin spacesIn chapter 3,firstly we list out several maximal operators.Secondly,we devote to the boundedness of the bilinear square operators associated with non-smooth kernels and their commutators.And lastly,we get some boundedness of bilinear Marcinkiewicz integral and bilinear Littlewood-Paley g-function,and the boundedness of the bilinear square operators and their commutators on the product of Morrey spaces will be given.In chapter 4,we summarize the main work of the thesis,and give some suggestions and expectations to the research in future.