A local regularity for the complex Monge-Amp\`ere equation

We prove a local regularity (and a corresponding a priori estmate) for plurisubharmonic solutions of the nondegenerate complex Monge-Amp'ere equation assuming that their $W^{2,p}$-norm is under control for some $p>n(n-1)$. This condition is optimal. We use in particular some methods developed by Trudinger and an $L^q$-estimate for the complex Monge-Amp'ere equation due to Ko{\l}odziej.