Some geometric properties related to smoothness of Banach spaces

In this paper, we introduce a new class of Banach spaces, locally kk nearly uniformly smooth spaces, which is the dual notion to locally kk nearly uniformly convex spaces introduced by Denka Kutzarova. The relationships between locally kk nearly uniformly smooth spaces, locally kk uniformly smooth Banach spaces, and locally fully kk smooth Banach spaces are discussed. We also introduce the two new smooth properties, (Sk-I) and (Sk-II), which are the extensions of smooth properties (S-I) and (S-II) that were introduced by Suyalatu, and discuss the relations between (Sk-I), (Sk-II) and other smooth properties. At last, the locally fully kk-smoothness, compact locally fully kk-smoothness, smooth properties (Sk-I) and (Sk-II) in product space lp(Xi)lp(Xi) are discussed.