Some Studies On Zygmund Type Spaces Weighted Operators

About analytic functions and harmonic functions smoothness of the boundary value of a function,Zygmund obtained the corresponding theorem.Later,many scholars analogya ?_? function,get Zygmund class functions,Zygmund theorems obtained a series of promotion,increase sexual function and the corresponding Zygmund space of analytic functions and smoothness.some people on this basis portray ?-Zygmund equivalent conditions in the unit disc for the study of nature and space ?-Zygmund type spaces provide a theoretical basis operator.The research of composition operator is the product of combination of the analytic function theory and operator theory operator theory,people have paid much attention to the properties operators and applications of research of operator since1960 s.Shortly people on the operator properties and applications of research focuses on increasing,growing interest,many scholars Composition Operators again extended to the weighted composition operator herein,this master's thesis focused on studying study author during Masters thesis,as well as some of the conclusions prove studied discussed Zygmund type spaces on the unit ball?small Zygmund type spaces?Operators on weighted boundedness and compactness characteristics,the main use of operator,the definition of the nature and function space on the norm,a simple test to select the appropriate function,get some bounded holomorphic function in space and compactness of several necessary and sufficient conditions article from the chapters to be discussed.In the first chapter,First,a brief description of the background research and domestic and international research status of Zygmund class functions,followed describe operator,generalized composite operator and weighted composition operator research background,development history and research value and significance,and finally the text commonly used in the later chapters basic concepts,mathematical symbols,and the normal number for a simple introduction.In the scend chapter,in the condition |f?x+h?-2f?x?+f?x-h?|?Ah, the Hardy-Littlewood theorem and Lipschitz conditions defined Zygmund type spaces ?_* functions,we definte ?_*^p function continuous integrationmode,?_*^p?t?=O?t?,???.Analogy to define ?_* function we definit ?_**^p function,if there is two positive number h₁,h₂,if h₁-h₂=??h??1/h??,h=max{h₁,h₂}is right,then ??? is also right.At the same time research Zygmund space in the unit disc ??? is sufficient and necessary conditions for f?z?continuous and analytic and inference.In the third chapter,we study discussions on the unit disk D is weighted Zygmund type spaces on Cesaro Operators boundedness and compactness to T_fg from Z₀^p to Z₀^q give operator boundedness and compactness of several necessary and sufficient conditions to further promote the conclusion,to obtain a composite function operator T_fg from Z₀^p to Z₀^q is a bounded and compactness several sufficient and necessary condition.In the fourth chapter,Research units from Zygmund type disk space to the weighted composite Bloch type spaces bounded sex,using test function on the unit sphere has been composite operator is a bounded operator necessary and sufficient conditions,further enriching the Zygmund space weighted Composition Operators conclusions.