Improved Caffarelli-Kohn-Nirenberg Inequalities and Uncertainty Principle

TL;DR

This paper advances Caffarelli-Kohn-Nirenberg inequalities and uncertainty principles for complex and vector-valued functions, introducing a phase derivative concept and extending results to higher-dimensional spheres.

Contribution

It introduces a novel phase derivative for vector-valued functions and extends uncertainty principles to functions on spheres, building upon previous work.

Findings

Established improved inequalities for complex- and vector-valued functions.

Extended the uncertainty principle to functions on spheres $ ext{S}^n$ for $n \\geq 2$.

Introduced a new phase derivative concept for vector-valued functions.

Abstract

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 $\mathbb S^n,n\geq 2.$