$L^p$-Caffarelli-Kohn-Nirenberg inequalities and their stabilities

Abstract

We establish a general identity (Theorem 1.2) that implies both the $L^p$-Hardy identities and the $L^p$-Caffarelli-Kohn-Nirenberg identities (Theorems 1.3 and 1.4) and $L^p$-Hardy inequalities and the $L^p$-Caffarelli-Kohn-Nirenberg inequalities (Theorems 1.5 and 1.6)). Weighted $L^p$-Caffarelli-Kohn-Nirenberg inequalities with nonradial weights are also obtained. (Theorem 1.7). Our results provide simple interpretations to the sharp constants, as well as the existence and non-existence of the optimizers, of several $L^p$-Hardy and $L^p$-Caffarelli-Kohn-Nirenberg inequalities. As applications of our main results, we are able to establish stabilities of a class of $L^2$ and $L^p$-Caffarelli-Kohn-Nirenberg inequalities. (Theorems 1.8 and 1.9.) We also derive the best constants and explicit extremal functions for a large family of $L^2$ and $L^p$ Caffarelli-Kohn-Nirenberg inequalities. (Corollaries 1.1 and 1.2.)

21 pages