RESEARCH ON GEOMETRY OF BANACH AND FUNCTION SPACES AND INEQUALITIES

着重研究了Banach空间和函数空间的几何结构与一些范数不等式的关系。我们还研究了巴拿赫空间的ψ-直接和，这似乎很重要，因为我们可以很容易地从凸函数ψ构造出许多具有非l_p型范数的巴拿赫空间的例子。主要结果如下。关于Banach空间和函数空间的几何结构及范数不等式：(1)推广了强随机克拉克森不等式，并将该不等式成立的Banach空间刻画为p-均匀光滑的Banach空间，或具有相同常数的强p型的Banach空间。(2)引入了带权的hanner型不等式，并利用这些不等式刻画了2-一致光滑空间和2-一致凸空间。通过改变权值的位置，得到的不等式表征了p均匀光滑空间和q均匀凸空间。对于l_p -空间，确定了这些不等式中权值的最佳值。并给出了这些汉纳型不等式的对偶定理。我们在Banach空间中给出了一个尖锐三角形不等式及其逆。关于Banach空间的ψ直和：(1)我们刻画了Banach空间的ψ直和的以下性质：光滑性、弱近一致光滑性、WORTH性质、Schur性质等。作为应用，我们给出了几个Banach空间的例子，它们不是一致非平方的，但具有不动点的性质。(2)我们特别刻画了Banach空间的l_1-sum和l_∞-sum的一致非l^n_1-ness （b_n -凹凸性）。[4]在James常数、性质M与不动点性质的关系、Banach-Mazur距离与超自反性的关系等方面得到了一些结果。

Geometric structures of Banach and function spaces were investigated especially in relation with some norm inequalities. Also ψ-direct sums of Banach spaces were investigated, which seem important as we can easily construct many examples of Banach spaces with a non l_p-type norm from a convex function ψ. Major results are as follows.[1]On geometric structures of Banach and function spaces and norm inequalities :(1)We extended the strong random Clarkson inequality, and characterized the Banach spaces in which the resulting inequality is valid are those of p-uniform smooth, or equivalently those of strong type p with the same constants involved them.(2)We introduced Hanner-type inequalities with a weight and characterized 2-uniformly smooth, and 2-uniformly convex spaces with these inequalities. By changing the place of the weight, the resulting inequalities characterize p-uniformly smooth, and q-uniformly convex spaces. The best value of the weight in these inequalities were determined for L_p-spaces. A duality theorem for these Hanner-type inequalities was presented as well.[2]We presented a sharp triangle inequality and its reverse in a Banach space.[3]On ψ-direct sums of Banach spaces :(1)We characterized the following properties of ψ-direct sums of Banach spaces :smoothness, weak nearly uniform smoothness, WORTH property, Schur property etc. As an application we presented several examples of Banach spaces which are not uniformly non-square, but have the fixed point property.(2)We characterized uniform non l^n_1-ness (B_n-convexity) particularly for the l_1-sum and l_∞-sum of Banach spaces.[4]Some resuts were obatined on the James constant, on the relation between Property M and the fixed point property, and on the Banach-Mazur distance in relation with super-reflexivity etc.