Research on Geometry and Probability in Banach Spaces and its Applications

从算符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一些推广，并利用这些不等式研究了Banach空间的泛函解析性质、几何性质和概率性质以及这些性质之间的关系。本文得到的主要结果如下：1。克拉克森型不等式和巴拿赫空间的几何：(1)从概率论的角度给出了一般巴拿赫空间设置下克拉克森不等式（CI）的高维版本。特别地，用Rademacher型和共型的概念给出了（CI）的表征。(2)证明了Lebesgue-Bochner空间L_r (X)如何从给定的Banach空间X继承clarkson型不等式。(3)利用clarkson型不等式表征了Banach空间的几何性质。2 . Banach空间中的Hanner， Hlawka和其他不等式：从概率论的角度给出了Hanner和Hlawka不等式的高维版本，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。3 . Banach空间的Von Neumann-Jordan （NJ-）常数与几何：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)，并用这些常数描述了X的几何性质。在巴拿赫空间理论和其他相关领域的应用：研究了泛函分析的一些与几何和概率性质有关的性质。研究了向量值遍历定理的一些应用。

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.