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和功能空间的几何结构，尤其是与Rademacher类型和Cotype的通知有关。还研究了BANACH空间的φ直接码头，这似乎很容易容易，可以从凸函数φ中构造许多具有非iptype norm的Banach空间的例子。主要结果如下。1。在几何结构和类型中，cotype：（1）我们介绍了强（rademacher）类型和cotype的音符，并表征了ρ-均匀平滑的表征，并用这些属性表达了Q-均匀的空间。这些特性对Lebesgue-Bochner空间L_γ（X）的遗产也显示出来。（2）我们引入了强烈的随机Clarkson不平等，并证明这种不平等在Banach空间中存在于Banach空间，并且仅当该空间具有强类型ρ，或者是相等的ρ-均匀平滑的。（回想一下，当且仅当空间具有ρ型； Kato-Persson-Takahashi，Collect。Math。51（2000）。）2。在几何结构和规范的不平等上：（1）汉纳型不等式通常可用于处理用Banach空间凸的模量描述的特性。我们考虑了汉纳（Hanner）型不平等，尤其是那些具有重量及其n个元素版本的不平等现象，并表征了最佳的2均匀凸度和均匀的非质量等。在banach空间的φ直接上：（1）我们表征了以下特性的Banach空间的φ-Direct总和：严格，均匀的凸度，均匀的非方面，非差异，非统一的非^n_1-性，反射性，反射性，弱几乎均匀的平稳性，平稳性，等于R2的绝对状态，该问题是ker n of Sove the r2的kal n offersightial n of So n of Sove to r2的问题，该问题是在R2上计算出的，该问题是按照R2的反应来计算的。（JMAA 258（2001））。