The complex Monge-Amp\`ere equation on weakly pseudoconvex domains

We show here a “weak” H"older-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Amp`{e}re equation with data in the $L^p$ space and the boundary of the domain satisfying an $f$-property. The $f$-property is a potential-theoretical condition which holds for all pseudoconvex domains of finite type and many examples of infinite type.