\quad F_t = \frac{\partial F}{\partial t}(v,z_2,\cdots,z_n,-z_1).
    \end{aligned}$$ When $\alpha=0$, by a similar computation we obtain with $\mathcal{F}_t={F_t}/{v_1}$.

A priori estimates
------------------

In this subsection, we obtain the a priori determinant estimates under the rotation transform $Z=TX$. Let $v$ be a smooth solution of satisfying $$\label{e219}
        v\geq0,\ \ v\geq z_1,\ \ v_1\geq0,$$ and $v(0)$ is as small as we want such that for the positive constant $s$ and $h$ in $(0,1/2)$, $\hat\Omega_{s,h}$ is a nonempty open set, where $$\label{e220}
        \hat\Omega_{s,h}=\{z\,:\,v(z)<sz_1+h\}.$$ Set $\hat v:=v-sz_1-h$, then $\hat\Omega_{s,h}=\{z\,:\,\hat v(z)<0\}$ and $\hat v$ satisfies $$\label{e221}
        \hat V^{ij}(\hat d^{\alpha-1})_{ij}=\hat g+\mathcal{\hat F}_t,$$ where $(\hat V^{ij})$ is the cofactor matrix of $(\hat v_{ij})$, $\hat d={\mbox{det}}\,D^2\hat v$ and $$\begin{aligned}
        \hat g\!\!&=&\!\!2\lambda(1-\alpha)\hat d^\alpha \frac{\hat v^{ij}\hat v_{ij1}}{\hat v_1+s}-(1-\alpha)(n+2)\lambda \hat d^\alpha\frac{\hat v_{11}}{(\hat v_1+s)^2}, \\
        \mathcal{\hat F}_t\!\!&=&\!\!\alpha^{-1} \frac{\hat F_t}{(\hat v_1+s)^\lambda}, \quad \hat F_t = \frac{\partial F}{\partial t}(\hat v+sz_1+h,z_2,\cdots,z_n,-z_1).
    \end{aligned}$$

\[l204\] Assume $0\leq\alpha\leq\frac{1}{n+2}$. Let $\hat v$ be a smooth solution of in $\hat\Omega_{s,h}$ and $\hat v=0$ on $\partial\hat\Omega_{s,h}$. Then for any $z\in\hat\Omega_{s,h}$, we have the a priori estimate $$\label{e222}
        {\mbox{det}}\,D^2\hat v\leq C,$$ where $C$ depends only $n,\alpha,{\mbox{dist}}(z,\partial\hat\Omega_{s,h}),\sup_{\hat\Omega_{s,h}}|\hat v|, \sup_{\hat\Omega_{s,h}}|D\hat v|$ and $\sup{\hat F}_t$.

When $\alpha=\frac{1}{n+2}$, the estimate easily follows from the affine invariant property. Note that in this case, $\lambda=0$ and $\hat g$ in 