H\"older regularity of the solution to the complex Monge-Amp\`ere equation with $L^p$ density

On a smooth domain $\mathrm{\Omega }\subset \subset {\mathbb{C}}^n$, we consider the Dirichlet problem for the complex Monge-Amp`ere equation $((d{d}^{c}u{)}^n=fdV,u{\mathrm{\mid }}_{b\mathrm{\Omega }}\equiv \varphi )$. We state the H"older regularity of the solution $u$ when the boundary value $\varphi$ is H"older continuous and the density $f$ is only $L^p$, $p>1$. Note that in former literature (Guedj-Kolodziej-Zeriahi) the weakness of the assumption $f\in L^p$ was balanced by taking $\varphi \in C^{1,1}$ (in addition to assuming $\mathrm{\Omega }$ strongly pseudoconvex).