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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)).

几何结构的Banach空间和功能的调查，特别是在关系的概念Rademacher型和cotype。本文还研究了Banach空间的φ-直和，这是一个很重要的问题，因为我们可以很容易地从一个凸函数φ构造出许多具有非i_p型范数的Banach空间的例子。主要结果如下：1.关于几何结构和型，余型：（1）引入了强（Rademacher）型和余型的概念，并利用这些性质刻画了ρ-一致光滑和q-一致凸空间。并证明了这些性质对Lebesgue-Bochner空间L_γ（X）的遗传性。(2)引入了强随机克拉克森不等式，证明了该不等式在Banach空间中成立当且仅当该空间是强ρ型的，或等价地是ρ-一致光滑的.（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ρ型; Kato-Clarkson-Takahashi，Collect.Math.51（2000）.）2.关于几何结构和范数不等式：（1）Hanner型不等式对于处理Banach空间的凸性模所描述的性质是有用的。研究了Hanner型不等式，特别是带权的Hanner型不等式及其n元形式，并利用这些不等式刻画了最优2-一致凸性和一致非方性等：（2）在Banach空间中引入Schaffer型常数，并给出了它与一致正规结构的关系.关于Banach空间的φ-直和：（1）刻划了Banach空间的φ-直和的下列性质：严格性、一致凸性、一致非平方性、一致非l_n_1性、自反性、弱近一致光滑性、光滑性等。(2)计算了R2上的绝对范数的James常数，部分回答了Kato-Ehranda（JMAA 258（2001））的一个问题.