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RESEARCH ON GEOMETRIC STRUCTURES OF BANACH AND FUNCTION SPACES AND APPLICATIONS

Banach几何结构与函数空间的研究及应用

基本信息

批准号：
14540181
负责人：
KATO Mikio
金额：
$ 2.3万
依托单位：
Kyushu 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：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540181/
关键词：
absolute norm Convex function φ-direct sum of Banach spaces sharp uniform convexity type-cotype strong type-cotype norm inequalitiy Banach spaces typ-cotype

项目摘要

Geometric structures of Banach and function spaces are investigated, especially in relation with the notions of Rademacher type and cotype. Also φ-direct sums of Banach spaces are investigated, which seem important as one can easily, construct a plenty of examples of Banach spaces with a non ι_ptype norm from a convex function φ. Major results are as follows.1.On geometric structures and type, cotype :(1) We introduced the notions of strong (Rademacher) type and cotype, and characterized ρ-uniformly smooth, and q-uniformly convex spaces with these properties. The heredity of these properties to Lebesgue-Bochner spaces L_γ(X) was shown, as well.(2) We introduced strong random Clarkson inequality, and proved that this inequality holds in a Banach space if and only if the space is of strong type ρ, or equivalently, ρ-uniformly smooth. (Recall that random Clarkson inequality holds in a Banach space if and only if the space has type ρ ; Kato-Persson-Takahashi, Collect. Math. 51 (2000).)2. On geometric structures and norm inequalities :(1) Hanner-type inequalities are often useful to treat properties described with the modulus of convexity of a Banach space. We considered Hanner-type inequalities, especially, those with a weight and their n element version, and characterized optimal 2-uniform convexity and uniform non-squareness, etc. with these inequalitites :(2) We introduced a Schaffer-type constant for a Banach space and showed a relation with uniform normal structure.3. On φ-direct sums of Banach spaces :(1) We characterized the following properties of a φ-direct sum of Banach spaces : strict, uniform convexity, uniform non-squareness, uniform non Ι^n_1-ness, reflexivity, weak nearly uniform smoothness, smoothness, etc.(2) The James constant of an absolute norm on R2 was calculated for some cases, which gives a partial answer to a problem of Kato-Maligranda (JMAA 258 (2001)).

研究了 Banach 和函数空间的几何结构，特别是与 Rademacher 类型和 cotype 概念的关系。还研究了巴纳赫空间的 φ 直和，这似乎很重要，因为我们可以很容易地从凸函数 φ 构造出具有非 ι_p 型范数的巴纳赫空间的大量示例。主要结果如下： 1.关于几何结构和类型、共型：(1)引入了强(Rademacher)型和共型的概念，并用这些性质刻画了ρ-均匀光滑空间和q-均匀凸空间。这些性质对 Lebesgue-Bochner 空间 L_γ(X) 的遗传性也得到了证明。 (2) 我们引入了强随机 Clarkson 不等式，并证明了当且仅当该空间是强类型 ρ 或等效地 ρ 均匀光滑时，该不等式在 Banach 空间中成立。（回想一下，随机克拉克森不等式在 Banach 空间中成立当且仅当该空间具有类型 ρ 时；Kato-Persson-Takahashi, Collect. Math. 51 (2000)。）2。关于几何结构和范数不等式：(1) Hanner 型不等式通常可用于处理用 Banach 空间的凸性模量描述的属性。我们考虑了汉纳型不等式，特别是那些带有权重及其n元版本的不等式，并用这些不等式表征了最优2-均匀凸性和均匀非方性等：（2）我们引入了Banach空间的Schaffer型常数，并展示了与均匀法式结构的关系。3．关于 Banach 空间的 φ 直和：（1）我们表征了 Banach 空间的 φ 直和的以下性质：严格、均匀凸性、均匀非正方形、均匀非 I^n_1 性、自反性、弱近均匀光滑性、光滑性等。（2）在某些情况下计算了 R2 上绝对范数的詹姆斯常数，这给出了以下问题的部分答案：加藤-Maligranda (JMAA 258 (2001))。