Szego projection versus potential theory for non-smooth planar domains

We show that the Kerzman-Stein operator associated to a bounded planar domain Ω with C¹-boundary is compact in L²(bΩ). We establish the Kerzman-Stein equation for the Szego projection associated to a bounded planar domain with Lipschitz boundary. As an application, we extend to the Lipschitz setting a theorem of S. Bell for representing the solution of the classical Dirichlet problem on a simply connected bounded domain in the complex plane.