【作者】 陈伟；

【导师】 缪爽； 尹万科；

【作者基本信息】 武汉大学 ， 基础数学， 2023， 博士

【摘要】 本学位论文主要研究多复变中的不变量问题与全纯函数空间.在不变量问题中,我们研究了拟凸域边界上的有限型和全纯映射的退化秩这两类不变量.对于拟凸域边界上的有限型,我们主要研究由Bloom-Graham引入的正则多重有限型,即交换子型、Levi型以及正则切触型,并且在光滑的拟凸实超曲面上,我们讨论了这三者之间的等价关系.对于全纯映射的退化秩,我们将退化秩的概念应用到全纯映射的刚性问题中,得到了全纯Segre映射的刚性性质.对于全纯函数空间,我们主要在Fock空间上讨论广义Cesàro算子以及径向导数算子的有界性和紧性.全文共分为四章.在第一章中,我们分别简要介绍了不变量问题与全纯函数空间的研究背景及现状,并给出了本学位论文的主要结果.在第二章中,我们研究了光滑拟凸实超曲面M∈Cⁿ⁺¹上的正则多重有限型,即交换子型t^（s）（M,p）、Levi型c^（s）（M,p）以及正则切触型a^（s）（M,p）,其中p∈M以及1≤s≤n.首先,我们简要介绍了权坐标系统的概念.借此,我们在Levi形式至少有n-k个正特征根的光滑拟凸实超曲面M上,证明了当s∈[k,n]时,上述三类有限型是相互等价的.作为推论,我们在Levi形式至少有n-1个正特征根的光滑拟凸实超曲面M上证明了Bloom猜想和D’Angelo猜想是成立的.在第三章中,我们以Heisenberg超曲面Hⁿ的复化为研究对象,主要研究Hⁿ到H^N（N≥n）上全纯Segre映射的刚性问题.我们引入了退化秩的概念,并且证明了当全纯Segre映射的退化秩为1时,全纯Segre映射具有刚性性质.在第四章中,我们以Fock空间为研究对象,其中0<m,p<+∞.在文献[2]的工作基础之上,借助F_m^p上的范数等价公式:我们完整刻画了广义Cesàro算子T_g:F_m^p→F_m^q和径向导数算子R:F_m^p→F_m^q是有界算子（或紧算子）的特征,其中0<p,q<+∞以及m>0.更多还原

【Abstract】 In this thesis,we mainly study the invariant problems and holomorphic function spaces in several complex variables.In the invariant problems,we study two types of invariants:finite types on the boundary of the pseudoconvex domain and the degenerate rank of holomorphic mappings.For finite types on the boundary of the pseudoconvex domain,we mainly study the regular multiple finite types introduced by Bloom-Graham,namely the commutator type,the Levi-form type and the regular contact type,and on the smooth and pseudoconvex real hypersurface,we discuss the equivalent relations among the three.For the degenerate rank of holomorphic mappings,we apply the concept of the degenerate rank to the rigidity problem of holomorphic mappings,and obtain rigidity properties of the holomorphic Segre mapping.For holomorphic function spaces,we mainly discuss the boundedness and compactness of the generalized Cesàro operator and the radial derivative operator on Fock space.This thesis is divided into four chapters.In the first chapter,we briefly introduce the research background and current situation of the invariant problems and holomorphic funcational spaces,and give the results of this thesis.In the second chapter,we study the regular multiple finite types on the smooth and pseu-doconvex real hypersurface M∈Cⁿ⁺¹,namely the commutator type t^（s）（M,p）,the Levi-form type c^（s）（M,p）and the regular contact type a^（s）（M,p）,where p∈M and 1≤s≤n.Firstly,we briefly introduce the concept of the weighted coordinate system.In this way,we prove that when s∈[k,n],the three finite types are mutually equivalent on the smooth and pseudoconvex real hypersurface M whose Levi form has at least n-k positive eigenvalues.As a corollary,we prove that the Bloom conjecture and the D’Angelo conjecture are valid on the smooth and pseudoconvex real hypersurface M whose Levi form has at least n-1positive eigenvalues.In the third chapter,we take the complexificationof the Heisenberg hypersurface Hⁿas the research object,and^j=m¹ainly study the rigidity prob-lem of the holomorphic Segre mapping from Hⁿinto H^N（N≥n）.We introduce the concept of the degenerate rank,and prove that when the degenerate rank of the holomorphic Segre mapping is 1,the holomorphic Segre mapping possesses rigidity properties.In the fourth chapter,we take the Fock spaceas the research object,where 0<m,p<+∞.Based on document[2],we use the following equivalent fo∫rmulap∫e-p|z|mon F_m^pto completely describe the characteristics of the generalized Cesàro operator T_g:F_m^p→F_m^qand the radial derivative operator R:F_m^p→F_m^qas bounded operators（or compact operators）,where 0<p,q<+∞and m>0.更多还原