The stability on the Caffarelli-Kohn-Nirenberg and Hardy-type inequalities and beyond
Yuxuan Zhou, Wenming Zou
TL;DR
This work establishes sharp gradient-stability results for the Caffarelli–Kohn–Nirenberg inequality in full parameter range, including functional and critical-point settings, by leveraging a simple transformation that reduces weighted to unweighted forms and yields optimal exponents. It further derives improved Sobolev-type embeddings with weak Lebesgue norms on general domains and in critical regimes, clarifying how domain geometry and spectral data influence stability constants. In the Hardy-type arena, the authors present weighted Hardy, logarithmic Sobolev, logarithmic Hardy, Hardy–Morrey, and interpolation inequalities, providing explicit stability constants and complete extremal classifications, with several results extended to convex cones and log-concave weights. Overall, the paper contributes explicit quantitative stability results and transformation-based methods that connect CK-N and Hardy-type inequalities, broadening applicability to weighted and domain-general settings. The findings have potential impact on analysis on weighted Sobolev spaces, extremal theory, and stability analysis in PDEs on conic and unbounded domains.
Abstract
In this paper, we establish several improved Caffarelli-Kohn-Nirenberg and Hardy-type inequalities. Our main results are divided into two parts. In the first part, we consider the following Caffarelli-Kohn-Nirenberg inequality: \begin{equation*} \left(\int_{\mathbb{R}^n}|x|^{-pa}|\nabla u|^pdx\right)^{\frac{1}{p}}\geq S(p,a,b)\left(\int_{\mathbb{R}^n}|x|^{-qb}|u|^qdx\right)^{\frac{1}{q}},\quad\forall\; u\in D_a^p(\mathbb{R}^n), \end{equation*} We establish gradient stability of this inequality in both functional and critical settings, and we derive some functional properties of the stability constant. Building on the gradient stability, we also obtain several refined Sobolev-type embeddings involving weak Lebesgue norms for functions supported in general domains. In the second part, we focus on various classical Hardy-type inequalities, including the standard Hardy inequality, the $L^p$-logarithmic Sobolev inequality with weights, the logarithmic Hardy inequality, the Hardy-Morrey inequality, the Hardy-Sobolev interpolation inequality, and the interpolated Caffarelli-Kohn-Nirenberg inequality. We investigate their weighted versions and derive corresponding extremal functions, refinements, new remaining terms and stability constants.