Let $u\in C^{2,\alpha}_0(\overline{\mathbb{R}^n_+})$, $0<\alpha<1$, such that
$$-\Delta u=f \quad\text{in }\quad \overline{\mathbb{R}^n_+}$$ $$ u=0 \quad\text{in }\quad \partial\overline{\mathbb{R}^n_+}$$
Then there exists a constant $C$ depending only on $N$ and $\alpha$ such that: $$[D^2u]_\alpha\leq C[f]_\alpha$$
where $$[u]_\alpha=\sup_{x,y\in\mathbb{R}^n_+}\frac{|u(x)-u(y)|}{|x-y|^\alpha}$$
I tried using Greens function for the halfspace and changing Morrays inequality demonstration but I'm stuck
Also, if $D^2u$ is a matrix what is the definition for $[D^2u]_\alpha$? i assumed its the maximum value of $[u_{x_ix_j}]_\alpha$ but im not sure about this.