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 $${\mathcal{L}}_{\varphi }u(x,t):=-u_t\mathrm{d}\mathrm{e}\mathrm{t}{D}^2\varphi (x)+\mathrm{t}\mathrm{r}[\mathrm{\Phi }(x)D^{2}u]=f(x,t)$$ under appropriate conditions on the domain, Monge-Amp$\grave{e}$re measures, boundary data and $f$, where $\mathrm{\Phi }:=\mathrm{d}\mathrm{e}\mathrm{t}(D^2\varphi )(D^2\varphi {)}^{-1}$ is the cofactor of the Hessian of $D^2\varphi$.