Caffarelli-Kohn-Nirenberg and Weighted Gaussian Poincar\'e Inequalities: a complete characterization of sharp $L^2$ stability and $L^p$ extensions

Abstract

We introduce a new family of weighted Gaussian $L^2$-Poincar\'e-type inequalities with explicit sharp constants, optimizers, and corresponding sharp $L^2$-gradient stability estimates. This family substantially extends the classical Gaussian Poincar\'e inequality. Owing to the singular nature of the weights involved, standard approaches to classical Gaussian Poincar\'e inequalities do not apply. To overcome this difficulty, we develop a new method based on generalized Laguerre polynomial expansions, spherical harmonic decompositions, and a Kelvin-type transform. As an application, we completely characterize the stability of the $L^2$-Caffarelli--Kohn--Nirenberg (CKN) inequalities by establishing sharp stability estimates, together with the stability of the stability inequality results, throughout the entire parameter range. Previous results were available only in a few special cases. We further establish weighted $L^p$-Poincar\'e inequalities for all $p>1$, and derive stability estimates for the $L^p$-CKN inequalities for $p\geq 2$ throughout the full parameter regime in which sharp constants and optimizers are known. In contrast, earlier $L^p$ results were restricted to highly limited parameter ranges.