  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.
