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arXiv · 2606.25692

The weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for the curl-free vector fields and second order derivatives: The sharp constants and stability estimates

Anh Tuan Duong·Van Hoang Nguyen
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Abstract

In this paper, we study the weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields and second order derivatives. Firstly, we prove a family of the sharp weighted second order $L^2$-Caffarelli-Kohn-Nirenberg inequalities that complements the results in [{\it C. Cazacu, J. Flynn and N. Lam, Calc. Var. Partial Differential Equations 62 (2023), no. 4, Paper No. 118, 26 pp.}] and [{\it A. T. Duong and V. H. Nguyen, On the sharp second order Caffarelli-Kohn-Nirenberg inequality. Ann. Fenn. Math., 50(1):275--286, 2025}]. Secondly, we establish a stability version of the sharp weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields proved by Cazacu, Flynn and Lam. Finally, we prove a stability estimate for the sharp weighted second order $L^2$-Caffarelli-Kohn-Nirenberg inequalities established in this paper. Our approach is based on the spherical harmonic decomposition method, the one dimensional integral inequalities and their improvements.

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Anh Tuan Duong, Van Hoang Nguyen. 2026-06-24. The weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for the curl-free vector fields and second order derivatives: The sharp constants and stability estimates. https://arxiv.org/abs/2606.25692

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