The second order Caffarelli-Kohn-Nirenberg identities and inequalities

Abstract

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 $N\ge1$, $t\ge p>1$, \begin{equation}\label{0.1} \left(\int_{\mathbb{R}^N} \frac{|\Delta u|^p}{|x|^{p\alpha}} \mathrm{d}x\right)^{\frac{1}{p}} \left[\int_{\mathbb{R}^N} \frac{\left|\nabla u\right|^{\frac{p(t-1)}{p-1}}} {|x|^{\frac{p(t-1)}{p-1}\beta}} \mathrm{d}x\right]^{\frac{p-1}{p}} \ge C(N,p,t,\alpha,\beta) \int_{\mathbb{R}^N} \frac{\left|\nabla u\right|^t}{|x|^{t\gamma}} \mathrm{d}x. \end{equation} Secondly, we establish second order $L^p$-Caffarelli-Kohn-Nirenberg identities, and obtain optimal constants and optimizers of the second order $L^p$-Caffarelli-Kohn-Nirenberg inequalities (i.e., $p=t$ in \eqref{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 \eqref{0.1} in radial space or in general space, and also establish the second order $L^p$-Caffarelli-Kohn-Nirenberg identities.