2024/05/11 by Chen, Xiao-Ping, Tang, Chun-Lei · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2405.06898
This paper focuses on optimal constants and optimizers of the second order Caffarelli-Kohn-Nirenberg inequalities. Firstly, we aim to study optimal constants and optimizers for the following second order Caffarelli-Kohn-Nirenberg inequality in radial space: let N≥1, t≥ p>1, (∫ℝN \frac|Δu|p|x|pα dx)(1)/(p) [∫ℝN \frac|∇ u|(p(t-1))/(p-1) |x|(p(t-1))/(p-1)β dx](p-1)/(p) ≥ C(N,p,t,α,β) ∫ℝN \frac|∇ u|t|x|tγ dx. Secondly, we establish second order Lp-Caffarelli-Kohn-Nirenberg identities, and obtain optimal constants and optimizers of the second order Lp-Caffarelli-Kohn-Nirenberg inequalities (i.e., p=t in \eqref0.1) in general space. Lastly, under some more general assumptions, we consider the optimal weighted second order Heisenberg Uncertainty Principles, which complements the recent work [``The sharp second order Caffareli-Kohn-Nirenberg inequality and stability estimates for the sharp second order uncertainty principle'', 2022, arXiv:2102.01425]. This paper's main novelty lies in the fact that we research the optimal versions of the second order Caffarelli-Kohn-Nirenberg inequalities \eqref0.1 in radial space or in general space, and also establish the second order Lp-Caffarelli-Kohn-Nirenberg identities.