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

Abstract

On a smooth domain $\Omega\subset\subset\mathbb C^n$, we consider the Dirichlet problem for the complex Monge-Amp\`ere equation $((dd^cu)^n=fdV,\,u|_{b\Omega}\equiv\phi)$. We state the H\"older regularity of the solution $u$ when the boundary value $\phi$ 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 $\phi\in C^{1,1}$ (in addition to assuming $\Omega$ strongly pseudoconvex).