Abstract
We establish Lp regularity for the Szego 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 Szego projection to be well defined, the local graph curve must be rectifiable). As applications, we obtain Lp 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.
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