Hausdorff Operators And Their Correlations

The Hausdorff operator was introduced to solve the convergence of the series by Hausdorff, and it is widely used in the fields of mathematics, such as harmonic analysis, complex analysis and partial differential equations, etc. Hausdorff op-erators contain the Hardy operator and its adjoint operator. Compared with Hardy operator, the boundedness of Hausdorff operators is far from perfect. So this paper is devoted to discuss the boundedness of the Hausdorff operators and their commutators on some function spaces, which extend some related results about Hardy operators. It is divided into six chapters.The first chapter is the preface, which introduce some results in this paper.The second chapter is preliminary knowledge. First we recall the definitions and properties of the Ap weights, and then some definitions of function spaces are given. Finally we introduce some operators and their relations.In Chapter Three, we give the weighted boundedness for commutators of high dimensional Hardy operators on Herz type spaces, including weighted Lip-schitz estimates and weighted CBMO estimates.In Chapter Four, we get CBMO estimates and Lipschitz estimates of com-mutator for fractional Hardy operator with a rough kernel on Herz type spaces. We also consider some estimates of generalized Hardy operator on Herz type spaces. Moreover, its λ-central BMO estimate on λ-central Morrey space is obtained.In Chapter Five, we discuss the boundedness of Hausdorff operators on some function spaces, and the some corresponding operator norms are worked out, which extend some related results about weighted Cesaro operator. Moreover, we use another method to prove the boundedness of n dimensional Hausdorff operator on Lesbesgue space.In Chapter Six, we study the boundedness of commutators of Hausdorff operators and fractional Hausdorff operators on Herz type spaces, including Lip- schitz boundedness and CBMO boundedness, and extend some results about commutators of Hardy operators.