Volterra型算子、面积积分算子的研究及其推广

项目名称： Volterra型算子、面积积分算子的研究及其推广

项目编号： No.11271162

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 马柏林

作者单位： 嘉兴学院

项目金额： 68万元

中文摘要： 解析函数 Hardy空间上的广义面积积分算子和 Volterra算子是两类重要算子,其有界性和紧性都可以用 Carleson型测度来刻画,关于它们的研究一直备受关注。将上述两类算子引入到定义在欧氏空间及其单位球面上的函数空间中，系统地研究它们在可积空间、实Hardy空间和有界振荡型空间等函数空间上的有界性和紧性特征，采用Carleson型测度来刻画有界性和紧性条件，是本项目主要研究的问题之一。由于离散的问题只能由差分的方法来讨论，将由导数定义的面积积分算子改变为由差分定义的更为广泛的（更有应用价值的）面积积分算子，研究新定义的算子的各种性质，是本项目主要研究的另外一个问题。综合应用调和分析和复分析的方法是本项目的一个特点，而用差分定义面积积分算子是一个全新的研究对象，因此，这些研究将扩展调和分析的研究方法和范围，极有可能带来方法和结果的创新。

中文关键词： 广义面积积分算子；Hardy型算子；加权估计；最佳常数；变指标可积空间

英文摘要： In the theory of Hardy spaces of analytic fucntions on the unit disc,the general area operators and Volterra operators are important ones and a lot of attention is paid to them.Their boundedness and compactness on Hardy spaces are characterized by Carleson measures.In the project, one of the main problems is to introduce area operators and Volterra operators to the function spaces defined on the Euclidean space and its unit sphere. We will plan to study the boundedness and the compactness of on integrabl function spaces,Hardy spaces and function spaces being of bounded mean oscillations by using Carleson measure. Since the data used may be given discretely, its varying is described by difference. The other problem is to introduce a more generalization of area operators by replacd the derivative by difference in the definition of the classical one and give the characterization of its boundedness and compactness on the function spaces above. In the project, we will apply comprehensively the methods used in harmonic analysis and analytic functions to study the problems. Since the area operator defined by difference is a new one, it may extend the range of study in harmonic analysis and is possible to give some new ideas from the research.

英文关键词： generalized area operator；Hardy operator；weighted estimates；best constant；integrable spaces with variable exponent