相关于高阶微分算子的函数空间实变理论及其应用

项目名称： 相关于高阶微分算子的函数空间实变理论及其应用

项目编号： No.11501506

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 曹军

作者单位： 浙江工业大学

项目金额： 18万元

中文摘要： 数学和物理中的很多问题往往可以归结为一些算子在相应函数空间中的有界性, 而为得到这些算子的有界性则需要对相应函数空间的实变理论做更深入的研究. 申请人与合作者已合作研究了一系列与各种二阶微分算子相关函数空间的实变理论以及相应Riesz变换的有界性, 并对部分高阶微分算子也得到了一些结果. 作为这些研究的继续和深入, 本课题拟进一步完善和发展与高阶微分算子相关函数空间的实变理论, 考虑与高阶齐次和非齐次椭圆算子相关的Hardy-Sobolev空间的实变理论, 研究与混合阶高阶椭圆微分算子相关的Hardy空间与Hardy-Sobolev空间的实变理论, 考虑相应Riesz变换在这些函数空间上的有界性质, 研究与高阶微分算子相关的Besov空间、Triebel-Lizorkin空间等函数空间的实变理论及其算子有界性, 并将由此得到的实变理论与Riesz变换有界性应用到相关方程解的正则性问题中.

中文关键词： 欧氏空间；高阶微分算子；函数空间；实变刻画；Riesz；变换

英文摘要： Many important problems arising in Mathematics and Physics can be reduced to the boundedness of some operators on the corresponding function spaces. To characterize these boundedness, one usually need to study the real variable theory of these function spaces. The applicant and his collarorators studied the real variable theory of the Hardy spaces associated with a series of second order differential operators, and obtained part results on some higher order differential operators. As a continuation and depth of these topics,this subject is devoted to complement and develop the real variable theory of function spaces associated with higher order differential operators. Moreover, this subject will intend to study the real variable theories of the Hardy-Sobolev spaces associated with higher order homogeneous and inhomogeneous elliptic operators, the Hardy and the Hardy-Sobolev spaces associated with higher order mixed order elliptic operators, consider the boundedness of the associated Riesz transform on these spaces and the corresponding Riesz transform characterization, introduce and study the real variable theory of the Besov spaces and Triebel-Lizorkin spaces associated with higher order differential operators and the boundedness of the associated Riesz transforms. Finally, this subject will apply all the obtained results to the study of the regularity properties of the associated differential equations.

英文关键词： Euclidean space;higher order differential operator;function space;real variable characterization;Riesz transform