The $L^2$-Norm of the Cauchy transform on circular annuli

TL;DR

This paper calculates the exact L^2 operator norm of the Cauchy transform on circular annuli, linking it to eigenvalues of a Laplacian problem with specific boundary conditions, and explicitly describes extremizers using Bessel functions.

Contribution

It provides the first exact computation of the L^2 norm of the Cauchy transform on annuli, connecting it to spectral properties of a Laplacian with mixed boundary conditions.

Findings

Exact L^2 operator norm expressed via Laplacian eigenvalues.

Reduction of the problem to a one-dimensional weighted Hardy operator.

Explicit description of extremizers using Bessel functions.

Abstract

We compute the exact $L^2$ operator norm of the Cauchy transform \[ (C_\Omega f)(z)=\frac1\pi\int_\Omega \frac{f(w)}{z-w}\,dA(w) \] on a circular annulus $\Omega=A(r,R)=\{r<|z|<R\}$. Exploiting rotational symmetry and a Fourier mode decomposition, we reduce the problem to a one--dimensional weighted Hardy operator and obtain \[ \|C_{A(r,R)}\|_{L^2\to L^2} = \frac{2}{\sqrt{\mu_1^{ND}(r,R)}}, \] where $\mu_1^{ND}(r,R)$ is the first eigenvalue of the Laplacian on $A(r,R)$ with Neumann condition on the inner boundary and Dirichlet condition on the outer boundary. The extremizers are explicitly described in terms of Bessel functions.