A new perspective on the Cauchy transform for non-smooth domains in the plane and applications

Loredana Lanzani, Purdue University

Abstract

We study the Kerzman-Stein operator in the setting of $C\sp1,$ 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 $L\sp2$-invertible on any bounded rectifiable domain in the complex plane whose Cauchy Transform is $L\sp2$-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 $C\sp1$ domain is compact. We obtain an application to potential theory which generalizes a result by S. Bell.

Degree

Ph.D.

Advisors

Bell, Purdue University.

Subject Area

Mathematics

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