Tomas Merchan

Kent State University

Talk Title: On the relation between L^2 boundedness and existence of principal value integral for a Calderón-Zygmund operator

Abstract: In 1998, Tolsa proved that any measure for which the Cauchy transform operator is bounded in L^2(\mu) also exists in the sense of principal value. However, it turns out that this is not the case in general. Jaye and Nazarov created a measure \mu in the complex numbers satisfying linear growth for which the singular integral operator with a simple kernel is bounded in L^2(\mu) but fails to exist in the sense of principal value. In the talk, we will introduce sharp sufficient conditions on a measure \mu which ensures that if a Calderón-Zygmund operator is bounded with respect to L^2(\mu), then the operator exists in the sense of principal value. This is joint work with Benjamin Jaye.