Quantitative stability for the Caffarelli-Kohn-Nirenberg inequality

Yuxuan Zhou, Wenming Zou

Published 2023-12-25Version 1

In this paper, we investigate the following Caffarelli-Kohn-Nirenberg inequality: \begin{equation*} \left(\int_{\mathcal{R}^n}|x|^{-pa}|\nabla u|^pdx\right)^{\frac{1}{p}}\geq S(p,a,b)\left(\int_{\mathcal{R}^n}|x|^{-qb}|u|^qdx\right)^{\frac{1}{q}},\quad\forall\; u\in D_a^p(\mathcal{R}^n), \end{equation*} where $S(p,a,b)$ is the sharp constant and $a,b,p,q$ satisfy the relations: \begin{equation*} 0\leq a<\frac{n-p}{p},\quad a\leq b<a+1,\quad 1<p<n,\quad q=\frac{np}{n-p(1+a-b)}. \end{equation*} Our main results involve establishing gradient stability within both the functional and the critical settings and deriving some qualitative properties for the stability constant. In the functional setting, the main method we apply is a simple but clever transformation inspired by Horiuchi \cite{Hor}, which enables us to reduce the inequality to some well-studied ones. Based on the gradient stability, we establish several refined Sobolev-type embeddings involving weak Lebesgue norms for functions supported in general domains. In the critical point setting, we use some careful expansion techniques motivated by Figalli and Neumayer \cite{Fig} to handle the nonlinearity appeared in the Euler-Lagrange equations. We believe that the ideas presented in this paper can be applied to treat other weighted Sobolev-type inequalities.