Szegö and Bergman projections on non-smooth planar domains

We establish L ^p regularity for the Szegö and Bergman projections associated to a simply connected planar domain in any of the following classes: vanishing chord arc; Lipschitz; Ahlfors-regular; or local graph (for the Szegö projection to be well defined, the local graph curve must be rectifiable). As applications, we obtain L ^p regularity for the Riesz transforms, as well as Sobolev space regularity for the non-homogeneous Dirichlet problem associated to any of the domains above and, more generally, to an arbitrary proper simply connected domain in the plane.