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 \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ , which is a further study of the results in Dang et al. (J Funct Anal 265:2239-2266, 2013). 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 \(\mathbb S^n,n\ge 2.\) ${\mathbb{S}}^n,n\ge 2.$