The Norm and Regularized Trace of the Cauchy Transform

In this paper, the norm of the Cauchy transform $C$ is obtained on the space $L^2(D,d\mu)$, where $d\mu=\omega(|z|)dA(z)$. Also, (for the case $\omega\equiv1$), the first regularized trace of the operator $C^*C$ on $L^2(\Omega)$ is obtained. The results are illustrated by examples, with different specific choices of the function $\omega$ and the domain $\Omega$.