Global $C^{1+\alpha,\frac{1+\alpha}{2}}$ regularity on the linearized parabolic Monge-Amp$\grave{e}$re equation

In this paper, we establish global $C^{1+\alpha,\frac{1+\alpha}{2}}$ estimates for solutions of the linearized parabolic Monge-Amp$\grave{e}$re equation $$\mathcal{L}_\phi u(x,t):=-u_t\,\mathrm{det}D^2\phi(x)+\mathrm{tr}[\Phi(x) D^2 u]=f(x,t)$$ under appropriate conditions on the domain, Monge-Amp$\grave{e}$re measures, boundary data and $f$, where $\Phi:=\mathrm{det}(D^2\phi)(D^2\phi)^{-1}$ is the cofactor of the Hessian of $D^2\phi$.