著名的Caffarelli-Kohn-Nirenberg不等式在数学形式上统一了众多早期著名的不等式，在泛函分析和经典的椭圆PDE研究中起到了关键性的作用。近年来，非局部椭圆型问题开始受到了广泛关注，由于在数学方法处理上，许多关于整数阶方程有效的方法对于分数阶方程模型来说却是完全失效的，相应更多关于分数阶的泛函不等式方面理论的建立和完善也显得越来越迫切。本项目拟建立起一类强大的分数阶插值不等式，就像CKN不等式在整数阶情形所扮演的角色一样，它也能统一起多种分数阶的泛函不等式。并对该不等式展开一些相关性质和应用的研究。本项目的研究方案是利用和发展一些新的理论和技巧，推广用于解决经典椭圆问题的变分与拓扑方法，使之适用于非局部问题的研究。分数阶版本的分部积分公式，Rellich 不等式，Pohozaev 恒等式，重排不等式，移动平面法等等将会被建立或改进。

The famous Caffarelli-Kohn-Nirenberg inequality includeds many early known inequalities as its special cases in mathmatical form, and plays a key role in functional analysis and classical ellipse PDE research. In recent years,the study of nonlocal equations of elliptic type has attracted substantial attention. Since so many methods about integer order equations are not valid for fractional equation model any more, it becomes urgent that more and more theories about the fractional order functional inequalities are needed to established. In this project, we shall establish a powerful fractional interpolation inequality, which can play a role, just like the role of CKN inequality in the integer order case, can also include many fractional functional inequalities as its special cases. In addition, some related properties and applications of the inequalities will be studied. The research methodology of this project is to develop and apply some new theories and techniques, to extend the variational and topological methods for solving classical elliptic problems, so that they can be applied to the study of nonlocal problems. Fractional version formula of integral by parts, Rellich inequalities, Pohozaev identities, rearrangement inequalities, moving plane methods and so on will be established or improved.