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

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.