If on a bounded smooth, pseudoconvex domain in $\mathbb{C}^n$, $\mathrm{det} ( \mathrm{Hess}(u)) = f$ ($f>0$, $\mathrm{Hess}(u)>0$, $u=0$ on the boundary), if $f \in C^{k, \alpha}$, is $u \in C^{k+2, \alpha}$ ? (I mean, is there an apriori estimate on $u$ with the Holder exponents of $f$ and $u$ being the same (equal to $\alpha$?)

If on a bounded smooth, pseudoconvex domain in $\mathbb{C}^n$, $\mathrm{det} ( \mathrm{Hess}(u)) = f$ ($f>0$, $\mathrm{Hess}(u)>0$, $u=0$ on the boundary), if $f \in C^{k, \alpha}$, is $u \in C^{k+2, \alpha}$ ? (I mean, is there an apriori estimate on $u$ with the Hölder exponents of $f$ and $u$ being the same (equal to $\alpha$?)