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

TL;DR

This paper investigates sharp L^2 bounds for the interior Cauchy transform on planar domains, revealing that previous conjectures are incorrect and establishing the precise constants, especially for simply connected domains and annuli.

Contribution

It identifies gaps in prior conjectures, constructs explicit counterexamples, and determines the exact sharp constants for the interior Cauchy transform in relation to the Dirichlet spectrum.

Findings

Counterexample on the unit disk shows the conjectured identity is false.

The correct sharp constant is identified in terms of a potential-type operator.

Testing on the first Dirichlet eigenfunction exceeds the spectral threshold, with equality only for disks.

Abstract

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.