Stability estimates for $L^p$-Caffarelli-Kohn-Nirenberg inequalities

TL;DR

This work advances the stability theory for $L^p$-Caffarelli-Kohn-Nirenberg inequalities by leveraging Figalli–Zhang vector inequalities and weighted Poincaré tools to establish quantitative stability results for both scale-invariant and scale-non-invariant forms. It resolves the $1<p<2$ stability question for the scale-invariant case by showing nonexistence of gradient-type stability and replacing it with a distance-to-extremizer framework, while providing robust stability bounds for $p\ge 2$ and for scale-non-invariant variants under precise parameter relations. A key methodological contribution is the derivation of weighted $L^p$-Poincaré inequalities on $\mathbb{R}^N$ and on log-concave Borel sets, which underpin the distance-based stability estimates toward the extremizer family $\mathcal{M}_{a,b}$. The results generalize prior $p=2$ findings and Do23’s work to the full range $1<p<N$, with potential applications to related stability questions and hydrogen-uncertainty principle variants.

Abstract

Based on some new vector inequalities established by Figalli and Zhang [\emph{Duke Math. J.} \textbf{171} (2022), 2407--2459], we study the stability of the scale invariant and the scale non-invariant $L^p$-Caffarelli-Kohn-Nirenberg inequalities, which fills the recent work of Do \emph{et al.} [$L^p$-Caffarelli-Kohn-Nirenberg inequalities and their stabilities, arXiv: 2310.07083] for $1<p<2$, and also extends some results of Cazacu \emph{et al.} [\emph{J. Math. Pures Appl. (9)} \textbf{182} (2024), 253--284] to a general case for $L^p$-Caffarelli-Kohn-Nirenberg inequalities with $1<p<N$.

Theorems & Definitions (30)

• Theorem A: Catrina09
• Theorem B: Do23
• Theorem C: Do23
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.5
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9