Caffarelli-Kohn-Nirenberg identities, inequalities and their stabilities

We set up a one-parameter family of inequalities that contains both the Hardy inequalities (when the parameter is 1) and the Caffarelli-Kohn-Nirenberg inequalities (when the parameter is optimal). Moreover, we study these results with the exact remainders to provide direct understandings to the sharp constants, as well as the existence and non-existence of the optimizers of the Hardy inequalities and Caffarelli-Kohn-Nirenberg inequalities. As an application of our identities, we establish some sharp versions with optimal constants and theirs attainability of the stability of the Heisenberg Uncertainty Principle and several stability results of the Caffarelli-Kohn-Nirenberg inequalities (see Theorem 1.1). In particular, for the stability of the Heisenberg uncertainty principle, we introduced a new deficit function ${\delta }_1\left(u\right)$ and established the best constant for the stability inequality (see Theorem 1.2) which also leads to the stability inequality with sharp constant with respect to the deficit function ${\delta }_2\left(u\right)$ and improved the known stability result for ${\delta }_2\left(u\right)$ in the literature. (see Theorem 1.3). A sharp stability inequality for the non-scale invariant Heisenberg uncertainty principle with optimal constant is also obtained in Theorem 1.4.