Research on Geometry and Probability in Banach Spaces and its Applications

Banach空间中的几何与概率研究及其应用

• 批准号: 09640214
• 负责人: TAKAHASHI Yasuji
• 金额: $ 1.92万
• 依托单位: Okayama Prefectural University
• 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-09640214/
• 关键词: Banach space; Rademacher type, cotype; uniformly convex space; Clarkson inequality; Neumann-Jordan constant; normal structure; norm inequalities; vector valued random variables; タイプ・コタイプ; ノイマン・ジョルダン定数; ラフカ不等式; 超回帰性; ルベ-グ・ボホナ-空間; 確率測度

We considered some generalizations of classical norm inequalities in general Banach spaces from the operator theoretical and probability theoretical point of view, and using these inequalities we investigated functional-analytic, geometric and probabilistic properties of Banach spaces as well as the relations between these properties.The main results obtained here are as follows.1. Clarkson-type inequalities and geometry of Banach spaces :(1) High-dimensional versions of the Clarkson inequality (CI) are given in the general Banach space setting from the probability theoretical point of view. In particular, a characterization of (CI) is given by the notions of Rademacher type and cotype.(2) It is shown how Clarkson-type inequalities are inherited by the Lebesgue-Bochner spaces L_r (X) from a given Banach space X.(3) Geometrical properties of Banach spaces are characterized using Clarkson-type inequalities.2. Hanner, Hlawka and the other inequlities in Banach spaces : High-dimensional versions of hanner and Hlawka inequalities are given from the probability theoretical point of view, and geometric and probabilistic properties of spaces are described by these inequalities and the others.3. Von Neumann-Jordan (NJ-) constant and geometry of Banach spaces : Some geometrical constants of Banach spaces X including NJ-constant C_<NJ>(X) and James constant J(X) are introduced, and geometrical properties of X are described in terms of these constants.4. Applications to Banach space theory and other related areas : Some properties of functional analysis are investigated in connection with geometric and probabilistic properties. Some applications to vector valued ergodic theorems are also investigated.

我们从算子理论和概率论的角度考虑了一般 Banach 空间中经典范数不等式的一些推广，并利用这些不等式我们研究了 Banach 空间的泛函分析、几何和概率性质以及这些性质之间的关系。主要研究结果如下： 1． Clarkson型不等式和Banach空间的几何：（1）从概率论的角度给出了一般Banach空间设置下的Clarkson不等式（CI）的高维版本。特别是，(CI) 的表征由 Rademacher 型和 cotype 的概念给出。(2) 展示了 Lebesgue-Bochner 空间 L_r (X) 如何从给定的 Banach 空间 X 继承 Clarkson 型不等式。 (3)利用Clarkson型不等式表征Banach空间的几何性质。2． Hanner、Hlawka 和 Banach 空间中的其他不等式：从概率论的角度给出了 hanner 和 Hlawka 不等式的高维版本，并通过这些不等式和其他不等式描述了空间的几何和概率性质。 3． Von Neumann-Jordan (NJ-) 常数和 Banach 空间的几何：介绍了 Banach 空间 X 的一些几何常数，包括 NJ-常数 C_<NJ>(X) 和 James 常数 J(X)，并描述了 X 的几何性质就这些常数而言。4。巴纳赫空间理论和其他相关领域的应用：结合几何和概率特性研究泛函分析的一些特性。还研究了向量值遍历定理的一些应用。

项目成果

Yasuji Takahashi and Mikio Kato: "The work of W.T.Gowers I-Presentation of the Fields Medalists" Sugaku(Iwanami). 51(2). 154-157 (1999)

高桥泰司和加藤干雄：“W.T.Gowers I 的作品——菲尔兹奖得主的颁奖”Sugaku（岩波）。

高橋泰嗣: "On minimal Banach spaces" 北海道大学数学講究録. 57. 22-25 (1999)

高桥靖：“关于最小巴纳赫空间”北海道大学数学Kokyuroku 57. 22-25 (1999)。

高橋泰嗣: "Recent Progress in Banach Space theory W.T.Gowersの業績をめぐって" 数理解析研究所講究録. to appear. (1999)

高桥靖：“关于 W.T. 高尔斯的成就的巴拿赫空间理论的最新进展”，数学科学研究所出版（1999 年）。

Yasuji Takahashi and Mikio Kato: "Von Neumann-Jordan constant and normal structure of Banach spaces" Proc.of the Real Analysis Symposium. 1997. 112-116 (1998)

Yasuji Takahashi 和 Mikio Kato：“Von Neumann-Jordan 常数和 Banach 空间的正规结构”Proc.of the Real Analysis Symposium。

Sin-Ei Takahasi, Yoshiaki Okazaki and Yasuji Takahashi: "A Hlawka type inequality and its converse" Hokkaido Univ.Technical Report Ser.in Math.52. 42-45 (1998)

Sin-Ei Takahasi、Yoshiaki Okazaki 和 Yasuji Takahashi：“Hlawka 型不等式及其逆”北海道大学技术报告系列数学 52。