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

We compute the exact $L^2$ operator norm of the Cauchy transform $$(C_{\mathrm{\Omega }}f)(z)=\frac{1}{\pi }{\int }_{\mathrm{\Omega }}\frac{f(w)}{z-w}dA(w)$$ on a circular annulus $\mathrm{\Omega }=A(r,R)=\{r<\mathrm{\mid }z\mathrm{\mid }<R\}$. Exploiting rotational symmetry and a Fourier mode decomposition, we reduce the problem to a one–dimensional weighted Hardy operator and obtain $$\mathrm{\parallel }{C}_{A(r,R)}{\mathrm{\parallel }}_{L^2\to L^2}=\frac{2}{\sqrt{{\mu }_1^{ND}(r,R)}},$$ 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.