A strong-form stability for a class of $L^p$ Caffarelli-Kohn-Nirenberg interpolation inequality

TL;DR

This paper proves a strong-form stability estimate for a class of Caffarelli-Kohn-Nirenberg interpolation inequalities in $L^p$ spaces, including second-order cases for radial functions, highlighting the relationship between deficits and function proximity.

Contribution

It establishes a novel strong-form stability result for CKN inequalities, including second-order cases, which was previously unknown.

Findings

Proved a stability inequality with explicit exponents for the CKN class.

Showed stability cannot be separately established for individual norms.

Extended results to second-order inequalities for radial functions.

Abstract

We study the stability of a class of Caffarelli-Kohn-Nirenberg (CKN) interpolation inequality and establish a strong-form stability as following: \begin{equation*} \inf_{v\in\mathcal{M}_{p,a,b}}\frac{ \|u-v\|_{H_b^p} \|u-v\|_{L^p_a}^{p-1} }{\|u\|_{H^p_b}\|u\|_{L^p_a}^{p-1}} \le C\delta_{p,a,b}(u)^{t}, \end{equation*} where $t=1$ for $p=2$ and $t=\frac{1}{p}$ for $p > 2$, and $\delta_{p,a,b}(u)$ is deficit of the CKN. We also note that it is impossible to establish stability results for $\|\cdot\|_{H_b^p}$ or $\|\cdot\|_{L_a^p}$ separately. Moreover, we consider the second-order CKN inequalities and establish similar results for radial functions.