Improved Caffarelli-Kohn-Nirenberg Inequalities and Uncertainty Principle

In this paper we prove some improved Caffarelli-Kohn-Nirenberg inequalities and uncertainty principle for complex- and vector-valued functions on ${\mathbb{R}}^n$, which is a further study of the results in \cite{Dang-Deng-Qian}. In particular, we introduce an analogue of “phase derivative” for vector-valued functions. Moreover, using the introduced “phase derivative”, we extend the extra-strong uncertainty principle to cases for complex- and vector-valued functions defined on ${\mathbb{S}}^n,n\ge 2.$