| We study the Kerzman-Stein operator in the setting of {dollar}Csp1,{dollar} Lipschitz, and less than Lipschitz domains in the plane. We extend a theorem by N. Kerzman and E. Stein by proving that the identity minus the so-called Kerzman-Stein operator is {dollar}Lsp2{dollar}-invertible on any bounded rectifiable domain in the complex plane whose Cauchy Transform is {dollar}Lsp2{dollar}-bounded. It follows that the equation of Kerzman and Stein for the Szego projection extends to the class of Lipschitz domains and, more generally, to a certain class of Ahlfors- regular domains. We show that the Kerzman-Stein operator associated to a {dollar}Csp1{dollar} domain is compact. We obtain an application to potential theory which generalizes a result by S. Bell. |
