Caffarelli-Kohn-Nirenberg identities, inequalities and their stabilities

2022/11/26 by Cristian Cazacu, Joshua Flynn, Cazacu, Cristian +5 · 5 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2211.14622

openalex publication_date 2022/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We set up a one-parameter family of inequalities that contains both the Hardy inequalities (when the parameter is 1) and the Caffarelli-Kohn-Nirenberg inequalities (when the parameter is optimal). Moreover, we study these results with the exact remainders to provide direct understandings to the sharp constants, as well as the existence and non-existence of the optimizers of the Hardy inequalities and Caffarelli-Kohn-Nirenberg inequalities. As an application of our identities, we establish some sharp versions with optimal constants and theirs attainability of the stability of the Heisenberg Uncertainty Principle and several stability results of the Caffarelli-Kohn-Nirenberg inequalities.

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• Sharp remainder terms and stability of weighted Hardy-Poincaré and Heisenberg-Pauli-Weyl inequalities related to the Baouendi-Grushin operator
• On the extremal functions of second order uncertainty principles: symmetry and symmetry breaking

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