与微分算子相关的加权Hardy型空间实变理论及应用

项目名称： 与微分算子相关的加权Hardy型空间实变理论及应用

项目编号： No.11471176

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 赵凯

作者单位： 青岛大学

项目金额： 70万元

中文摘要： Hardy空间实变理论和奇异积分算子理论构成了调和分析的核心内容，并与数理方程等领域有着密切联系。由于偏微分方程、复分析、几何分析的需要，与微分算子相关的奇异积分算子得到了长足发展，相关于微分算子的新型Hardy空间理论也十分活跃。近年来，申请人及合作者在多参数Hardy空间、各向异性的加权Hardy空间和Hardy型空间、乘积域上相关于算子的Hardy空间等函数空间上建立了原子或分子分解特征，得到了包括相关于Schr?dinger算子在内的一些奇异积分算子及交换子在函数空间的有界性等结果。本课题拟研究乘积域上相关于微分算子的加权Hardy空间的实变理论，主要包括原子刻画、极大函数刻画、对偶理论等。拟建立与微分算子相关的Herz型Hardy空间、及加权空间的实变理论。继而，讨论与微分算子相关的以Calderón-Zygmund算子为代表的奇异积分算子及交换子在这些Hardy型空间的有界性。

中文关键词： Hardy空间；微分算子；乘积域；实变理论；奇异积分算子

英文摘要： The real variable theory of the Hardy spaces and the theory of singular integral operators are the core content of harmonic analysis and closely related to equations of mathematical physics and other fields. Due to the need of partial differential equations, complex analysis and geometric analysis, singular integral operators associated with differential operators have been extensively development. Then the theory of new type of Hardy spaces associated with differential operators is very active. Recent years, we worked on the multiparameter Hardy spaces, anisotropic weighted Hardy spaces, anisotropic weighted Herz type Hardy spaces and Hardy spaces associated to operators on product domains, established the atomic or molecular decompositions. And we also obtained the boundedness of some singular integral operators and commutators, including some operators associated with Schr?dinger operators on some function spaces. In this project, we will study the real variable theory of the weighted Hardy spaces associated with operators on product domains, including the atomic decomposition, maximal characterizations and the theory of the duality. The real variable theory of the Herz type Hardy spaces associated with differential operators and the weighted Herz type Hardy spaces will be also established. Further, we will discuss the boundedness of Calderón-Zygmund operators and other singular integral operators and commutators associated with differential operators on the new type Hardy spaces.

英文关键词： Hardy space;differential operator;product domain;real variable theory;singular integral operator