Let $\Delta_{0}$ be the standard Hodge Laplacian on the Euclidean space $\mathbb{R}^{3}$, then the following holds:

1. (1)

   Let $\{y_{1},y_{2},y_{3}\}$ be the Cartesian coordinates of $\mathbb{R}^{3}$, then

   (3.160)

   $\displaystyle\begin{cases}\Delta_{0}r=-\frac{2}{r},\\ \Delta_{0}(\frac{y_{\alpha}y_{\beta}}{r})=\frac{4y_{\alpha}y_{\beta}}{r^{3}},&\alpha\neq\beta,\\ \Delta_{0}((\frac{y_{\alpha}^{2}}{r}-r))=\frac{4y_{\alpha}^{2}}{r^{3}},\\ \Delta_{0}(\frac{y_{\alpha}}{r})=\frac{2y_{\alpha}}{r^{3}}.\end{cases}$

2. (2)

   Denote by $\mathcal{P}_{4}$ the space of all homogeneous degree 4 polynomials on $\mathbb{R}^{3}$, then the operator

   (3.161)

   $$\square:\mathcal{P}_{4}\rightarrow\mathcal{P}_{4};f\mapsto r^{5}\Delta_{0}(r^{-3}f)$$

   is an isomorphism.