Studies of The Structures of Analytic Function Spaces and Their Operators

解析函数空间及其算子的结构研究

• 批准号: 09640218
• 负责人: OHNO Shuichi
• 金额: $ 1.98万
• 依托单位: Nippon Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640218/
• 关键词: Hardy Space; Composition Operator; Hopf hypersurface; Cayley algebra; Automorphism group; Meromorphic function; Complex functional equation; Monoid rings; 超越函数; スピノール群; 値分布論; 自由分解

(1) Ohno has investigatcd the 1)robleln of components of composition operators on H^* and obtained almost conIl)Iete answers with Professor IL Zhao. We had a talk at Poster Session in International Matlnnaticiazhs Goiigrcss 1998, Berlin, Garmany. After thc talk, wc summarized results to a paper with Profcssor beta.D). MacChicr and(l submitted it. Moreover Olino have studied weighted composition operators on some sup-norm function spaces, the disk algebra, H^* and the Bloch space. And R.Zha.o added results in the case of the case little Bloch space. We are ready to submit a manuscript.(2) Funabashmi studied time geometrical properties of the SP(1)-orbits which are realized by the special kinds of three actions to tIme nearly Kachler 6-sphere. We proved that each orbit is a manifold and somne orbit equips the contact CR-structure having the three distinct principal curvatures. Those results will be contributed in the imear future. We also studied about contact CR-submanifolds immersed in … More Sasakian space forms. Our maun results is that some totally contact Al-umbilical contact CR-submanifold is realized as the 2-dimensional torus immersed in the 3-dimensional sphere. For those results, we will contribute the paper entitled "On totally contact Al-umbilical contact CR-submanifolds" in collaboration with S.Funabashi, J.H.Kwon and J.S.Pak.(3) Let S^6 be the 6-dimensional unit sphere centered at the origin in a 7-dimensional Euclidean space. Hashimnoto identified 7-dimensional Euchidean space with purely imaginary octon ions ImO (or Cayley algebra). Taking account of algebraic properties of octonions we can define the homogeneous almost Itermitian structure on S^6, We denote by G_2 the Lie group of autornorphisms of O.Then we have S^6 = G_2/SU(3). This almost colnj)lex structure satisfy the nearly Kahler condition. ((*xJ)X = 0) where * is the Levi-Civita connection of S^6, and X is any vector field of S^6. We shall give some rigidity theorem of invariant submanifolds up to the action of G_2 amid deterirmine its geometrical invariants. Also, we shall give many examples of 3-dimensional CR-snbmanifolds of S^6 explicitely. We obtained some results related to 4-dimensional CR-submanifolds of S^6.(4) Ishmizaki has beemi studying the value distribution theory of meromorphic functions. Applications this theory to com np hex differential equations are of our interest. Algebraic differential equations admitting admissible solutiomis and complex oscillation theory have been comisidered. We are also concerned with functional equations in the complex plane. Results of existence and growth conditions on transcendental meromorphic solutions of Schmrdder's type functional equations, which are some generalizations due to Wittich, are obtained. Moreover, we investigated to lmypertranscendency of merornorphic solutiohs of a certain functional equation. Characterization of the set of meromorphic. functions has been studied from the unicity tlmeoretical poimits of view.(5) Eto investigated homnological properties of monoid rings, especially affine semigroup rings. To do it, lie comistructedI free resolntiomis of them in two cases conibinatorically. They are found in papers "a free resolutions of a binomial ideal" and "finite free resolutions of rnonoid rings". Less

(1) Ohno研究了H^*上复合算子的分量问题，并与赵立教授得到了几乎相同的答案。1998年，在德国柏林举行的国际数学联合会海报会上，我们进行了一次谈话。谈话结束后，我和贝塔教授一起将结果总结成一篇论文。MacChicr和我提交了它。此外，Olino还研究了一些超范数函数空间、磁盘代数、H^*和Bloch空间上的加权复合算子。r。zhao在小布洛赫空间的情况下添加了结果。我们已准备好投稿。(2) Funabashmi研究了SP(1)-轨道的时间几何性质，SP(1)-轨道是由近Kachler 6球对时间的三种特殊作用实现的。我们证明了每个轨道是一个流形，并且某些轨道具有三个不同主曲率的接触cr结构。这些成果将在不久的将来提交。我们还研究了接触cr -子流形浸入…更多的sasaki空间形式。我们的研究结果是，一些完全接触al -脐接触cr -子流形被实现为浸没在三维球体中的二维环面。对于这些结果，我们将与S.Funabashi， J.H.Kwon和J.S.Pak合作发表题为“关于完全接触al -脐带接触cr -子流形”的论文。(3)设S^6为7维欧几里德空间中以原点为中心的6维单位球。Hashimnoto用纯想象的八元离子（或Cayley代数）确定了7维欧几里德空间。考虑到八元数的代数性质，我们可以定义S^6上的齐次几乎迭代结构，用G_2表示0的自同态李群，得到S^6 = G_2/SU(3)。这种近似柯氏lex结构满足近似Kahler条件。（*xJ）X = 0)其中*为S^6的Levi-Civita连接，X为S^6的任意向量场。在确定G_2的几何不变量的同时，给出了G_2作用下不变子流形的刚性定理。此外，我们还将明确地给出S^6的三维cr -snb流形的许多例子。得到了S^6的四维cr -子流形的一些结果。(4) Ishmizaki开始研究亚纯函数的值分布理论。将这一理论应用于十六进制微分方程是我们感兴趣的。讨论了允许解的代数微分方程和复振荡理论。我们也关注复平面上的泛函方程。得到了schmrder型泛函方程的超越亚纯解的存在性和生长条件，这是wititch的一些推广。此外，我们还研究了一类泛函方程亚纯解的超超越性。亚纯集合的刻画。从唯一性气象学的角度对函数进行了研究。(5) Eto研究了单群环，特别是仿射半群环的同调性质。为了做到这一点，我在两种情况下构建了它们的自由解析。它们在论文“二项式理想的自由分辨”和“类环的有限自由分辨”中被发现。少

项目成果

K.Ishizaki(共): "Unicity theorems for meromorphic sharing four small functions" Koclai Muth. J.21. 350-371 (1998)

K.Ishizaki(co)：“亚纯共享四个小函数的唯一性定理”Koclai Muth J.21 (1998)。

S.Ohno: "Rocky Mountain Mathematis Consortium Summer Conference 1996 Composition Operators on spaces of analytic functins" Report of Reseanches N.I.T. 26. 421-436 (1997)

S.Ohno：“落基山数学联盟夏季会议 1996 年分析函数空间的组合算子”N.I.T 研究报告

K.Eto: "Cohen-Macaulay rings associated with digraphs" J.Algebra. 206. 541-544 (1998)

K.Eto：“与有向图相关的科恩-麦考利环”J.代数。

K.Izuchi(共): "Self adjoint commutators and invariant subapaces on the tows II" Integral Eguations and Operator Theory. 27. 208-220 (1997)

K. Izuchi（合著者）：“丝束 II 上的自伴随交换子和不变子空间”积分方程和算子理论 27. 208-220 (1997)。

T.Koda(共): "Grasamann Geometry of 6-dimensional sphere" Topics in Complex Aralysis,Differential geometry and mathematical Physics. (1997)

T.Koda（合作）：“六维球体的格拉萨曼几何”复数分析、微分几何和数学物理主题（1997）。