The $L^2$ Norm of the Interior Cauchy Transform: Beyond the First Dirichlet Eigenvalue

We study sharp $L^2$ bounds for the interior Cauchy transform $C_D$ on a bounded planar domain $D$ and clarify its connection with the Dirichlet spectrum. We analyze an approach that replaces fractional Dirichlet powers on $D$ by Euclidean Fourier multipliers after extension by zero, and show that this substitution can change the optimal constants. In particular, we construct an explicit endpoint counterexample on the unit disk to a Fourier-weighted inequality appearing in \cite{Dostanic1996}. This identifies a gap in the derivation of the conjectured identity $\|C_D\|_{L^2\to L^2}=2/\sqrt{\lambda_1(D)}$. We then identify the correct sharp constant for the endpoint Fourier weight $|\xi|^{-1}$ in terms of the top eigenvalue of a natural positive potential-type operator on $D$. Finally, we show that testing $C_D$ on the first Dirichlet eigenfunction already exceeds the spectral threshold, so $\|C_D\|_{L^2\to L^2}>2/\sqrt{\lambda_1(D)}$ for simply connected domains and also for annuli $A(r,R)$, and we prove a rigidity result: equality with the spectral value occurs if and only if $D$ is a disk.