The second order Caffarelli-Kohn-Nirenberg identities and inequalities

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, {equation}{0.1} (∫R^N {|Δ u|^p}{|x|pα} dx)1/p [∫R^N {|∇ u|p(t-1)/p-1} {|x|p(t-1)/p-1β} dx]p-1/p ≥ C(N,p,t,α,β) ∫R^N {|∇ u|^t}{|x|tγ} dx. {equation} 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 {0.1}) 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 {0.1} in radial space or in general space, and also establish the second order Lᵖ-Caffarelli-Kohn-Nirenberg identities.