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

1. (1)

   Denote by $\nu_{y}$ one of the following differential forms $dy_{\alpha}$, $dy_{\alpha}\wedge dy_{\beta}$ or $dy_{1}\wedge dy_{2}\wedge dy_{3}$. Similarly, let $\tau_{x}$ be a tangential $p$-form given by $\tau_{x}\equiv dx_{1}\wedge\ldots\wedge dx_{\alpha_{p}}$ with $0\leq p\leq m-3$. Let

   (3.164)

   $$\omega\equiv f(x)h(y)\nu_{y}\wedge\tau_{x},$$

   where $f(x)$ is a smooth function defined on $U$ and $h(y)=O^{\prime}(|y|^{k})$ for some $k\in\mathbb{Z}_{+}$, then

   (3.165)

   $$\Delta\omega-f(x)\cdot\Delta_{0}(h(y))\cdot\nu_{y}\wedge\tau_{x}=O^{\prime}(|y|^{k-1}).$$

2. (2)

   Let $f$ be a smooth function defined on $U\subset P$ and let

   (3.166)

   $$\omega\equiv f(x)\cdot\frac{y_{\alpha}}{r}dy_{1}\wedge dy_{2}\wedge dy_{3},$$

   then

   (3.167)

   $\displaystyle\Delta\omega=2r^{-2}\omega+r^{-5}\Gamma_{3}^{(4)}+O^{\prime}(1),$

   where the definition of the $3$-form $\Gamma_{3}^{(4)}$ is in (3.45) of Notation 3.9.