Suppose u is a smooth strictly plurisubharmonic function satisfying $(i\partial \bar{\partial} u)^n =f dV_{Euc}$ on the unit ball centered at the origin, where $f>0$ is a smooth function. Let $C$ be a constant such that $\Vert f \Vert_{C^{0,a}}\leq C$. Is some norm of $u$ over the half-ball controlled by $C$?

Suppose u is a smooth strictly plurisubharmonic function satisfying $(i\partial \bar{\partial} u)^n =f dV_{Euc}$ on the unit ball centered at the origin, where $f>0$ is a smooth function. Let $C$ be a constant such that $\Vert f \Vert_{C^{1,a}}\leq C$. Is some norm of $u$ over the half-ball controlled by $C$? Note that there is no specified boundary condition