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.

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推广，利用这些不等式研究了Banach空间的泛函-解析、几何和概率性质以及这些性质之间的关系，主要结果如下. Clarkson型不等式与Banach空间几何：（1）在一般Banach空间中，从概率论的角度给出了克拉克森不等式（CI）的高维形式.特别地，通过Rademacher型和cotype的概念给出了（CI）的特征。(2)本文证明了Clarkson型不等式是如何从给定的Banach空间X继承到Lebesgue-Bochner空间L_r（X）的. (3)利用Clarkson型不等式刻画了Banach空间的几何性质. Banach空间中的Hanner，Hlawka等不等式：从概率论的角度给出了高维形式的Hanner和Hlawka不等式，并用这些不等式和其他不等式描述了空间的几何和概率性质. Von Neumann-Jordan（NJ-）常数与Banach空间的几何：引入了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），并利用这些常数描述了X的几何性质.应用到Banach空间理论和其他相关领域：一些功能分析的性质进行了调查，与几何和概率性质。一些应用向量值遍历定理也进行了研究。

项目成果

高橋泰嗣: "Remarks on Williams-Wells′ inequality" 北海道大学数学講究録. 52. 46-49 (1998)

高桥靖：《关于威廉姆斯-威尔斯不等式的评论》北海道大学数学科急录 52. 46-49 (1998)

Aoi Honda, Yoshiaki Okazaki and Yasuji Takahashi: "An extension of Hlawka's inequality" RIMS Kokyuroku. l039. 12-19 (1998)

Aoi Honda、Yoshiaki Okazaki 和 Yasuji Takahashi：“Hlawka 不平等的延伸”RIMS Kokyuroku。

Yasuji Takahashi and Mikio Kato: "Geometry of Banach soaces and norms of*1 matrices" RIMS Kokyuroku. 1039. 165-169 (1998)

Yasuji Takahashi 和 Mikio Kato：“Banach soaces 的几何和*1 矩阵的范数”RIMS Kokyuroku。

Yasuji Takahashi and Mikio Kato: "Von Neumann-Jordan constant and uniformly non-square Banach spaces" Nihonkai Math.J.9(2). 155-169 (1998)

Yasuji Takahashi 和 Mikio Kato：“Von Neumann-Jordan 常数和均匀非方形 Banach 空间”Nihonkai Math.J.9(2)。

Yasuji Takahashi and Mikio Kato: "On minimal Banach spaces" Hokkaido Univ.Technical Report Ser.in Math.57. 22-25 (1999)

Yasuji Takahashi 和 Mikio Kato：“关于最小 Banach 空间”北海道大学技术报告系列数学 57。