RESEARCH ON GEOMETRIC STRUCTURES OF BANACH AND FUNCTION SPACES AND APPLICATIONS

研究了Banach空间和函数空间的几何结构，特别是与Rademacher型和余型概念的关系。还研究了Banach空间的φ直和，这似乎很重要，因为人们可以很容易地从凸函数ι构造出大量具有非φ_p型范数的Banach空间的例子。主要结果如下：1.关于几何结构和类型、余型：(1)我们引入了强(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。(2)引入了强随机Clarkson不等式，证明了该不等式在γ空间中成立的充要条件是该空间是强型ρ或等价的ρ一致光滑的。(回想一下，随机克拉克森不等式在Banach空间中成立的充要条件是该空间的类型为ρ；数学课。51(2000).)2.关于几何结构和范数不等式：(1)对于用Banach空间的凸性模描述的性质，汉纳型不等式通常是有用的。我们考虑了HANNER-型不等式，特别是具有权的HANNER-型不等式及其n元形式，并用这些不等式刻画了最优的2-一致凸性和一致非方形等：(2)引入了Banach空间的Schaffer型常数，并证明了它与一致正规结构的关系。关于Banach空间的φ-直和：(1)刻画了Banach空间的φ-直和的严格、一致凸性、一致非方形、一致非Ι^n_1-性、自反性、弱近一致光滑性、光滑性等性质.(2)在某些情况下，计算了R2上绝对范数的James常数，部分回答了Kato-Maliganda(Jmaa258(2001))的一个问题.

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)).