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

TL;DR

This paper establishes stability estimates for $L^p$-Caffarelli-Kohn-Nirenberg inequalities, extending recent results and filling gaps for different ranges of p and general cases.

Contribution

It provides new stability estimates for $L^p$-Caffarelli-Kohn-Nirenberg inequalities, expanding the understanding for various p ranges and non-invariant cases.

Findings

Fills recent gaps for 1<p<2

Extends results to general p in (1,N)

Provides new vector inequalities for stability analysis

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 $1<p<N$.