The homogeneous equation (5.17) was studied by the authors in the special case $n=\dim_{\mathbb{C}}(\mathcal{C}^{n})=2$. When $j_{k}=0$, (5.17) has standard solutions given by exponential functions. When $j_{k}>0$, the transformation was chosen as

(5.28)

$$\begin{cases}y=j_{k}^{\frac{1}{2}}\cdot z^{\frac{n}{2}}\\ u_{k}(z)=e^{-\frac{j_{k}z^{n}}{2}}\cdot Q(j_{k}^{\frac{1}{2}}\cdot z^{\frac{n}{2}}).\end{cases}$$

We refer the readers to Section 4 of [HSVZ18] for more details. In the special case $n=2$, $Q(y)$ is an Hermite function which satisfies the Hermite differential equation

(5.29)

$$\frac{d^{2}Q(y)}{dy^{2}}-2y\frac{dQ(y)}{dy}-2(h+1)Q(y)=0.$$

The key tool to prove the estimates for $Q$ essentially relies on its integral representation formula. However, when $n>2$, if we perform the transformation as (5.28) then the resulting equation for $Q$ is more complicated to study. It turns out the transformation (5.25) is a more suitable choice.