The L2‐norm of the Cauchy transform on circular annuli

Abstract

We compute the exact operator norm of the Cauchy transform on a circular annulus . Exploiting rotational symmetry and a Fourier mode decomposition, we reduce the problem to a one‐dimensional weighted Hardy operator and obtain where is the first eigenvalue of the Laplacian on with Neumann condition on the inner boundary and Dirichlet condition on the outer boundary. The extremizing modes are explicitly described in terms of Bessel functions.