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

TL;DR

This paper develops a unified framework for $L^{p}$-Hardy and $L^{p}$-Caffarelli-Kohn-Nirenberg inequalities, providing sharp constants, extremal functions, and stability results, with applications to nonradial weights and a broad class of inequalities.

Contribution

The authors introduce a general identity that unifies and extends existing inequalities, offering new insights into sharp constants, optimizers, and stability for $L^{p}$-Hardy and Caffarelli-Kohn-Nirenberg inequalities.

Findings

Established a general identity linking various $L^{p}$ inequalities.

Derived sharp constants and explicit extremal functions.

Proved stability results for a class of inequalities.

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.)