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

2023/10/10 by Do, Anh Xuan, Flynn, Joshua, Lam, Nguyen +1 · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.07083

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 L² 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 L² and L^p Caffarelli-Kohn-Nirenberg inequalities. (Corollaries 1.1 and 1.2.)

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