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

David Kalaj

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 \(\|C_D\|_{L^2\to L^2}=2/\sqrt{λ_1(D)}\). We then identify the correct sharp constant for the endpoint Fourier weight $|ξ|^{-1}$ 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 $\|C_D\|_{L^2\to L^2}>2/\sqrt{λ_1(D)}$ 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.

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

Abstract

We study sharp bounds for the interior Cauchy transform on a bounded planar domain and clarify its connection with the Dirichlet spectrum. We analyze an approach that replaces fractional Dirichlet powers on 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 \(\|C_D\|_{L^2\to L^2}=2/\sqrt{λ_1(D)}\). We then identify the correct sharp constant for the endpoint Fourier weight in terms of the top eigenvalue of a natural positive potential-type operator on . Finally, we show that testing on the first Dirichlet eigenfunction already exceeds the spectral threshold, so for simply connected domains and also for annuli , and we prove a rigidity result: equality with the spectral value occurs if and only if is a disk.
Paper Structure (13 sections, 6 theorems, 115 equations)

This paper contains 13 sections, 6 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Background, motivation, and main results
  3. A Fourier-multiplier pitfall on bounded domains
  4. A counterexample to the \ref{['eq:wrong-fourier-ineq']} at $\alpha=\tfrac{1}{2}$ and the correct sharp constant for the $\left|\xi\right|^{-1}$ Fourier weight
  5. Schur's test (used here).
  6. First-eigenfunction test and disk rigidity
  7. Boundary identity.
  8. Rigidity via the exterior conformal map.
  9. Competing interests.
  10. Data availability.
  11. Ethical approval.
  12. Consent to participate.
  13. Consent for publication.

Key Result

Theorem 1

Let $D\subset\mathbb{C}$ be a bounded domain with piecewise $C^1$ boundary. Then $C_D$ extends boundedly to $L^2(D)$ and where $\lambda_1(D)>0$ is the smallest eigenvalue of the Dirichlet problem

Theorems & Definitions (15)

  • Theorem 1: Claimed in Dostanic1996
  • Theorem 2: Sharp $L^2$ constant for the $|\xi|^{-1}$ Fourier weight
  • proof
  • Remark 3
  • Theorem 4
  • proof
  • Lemma 5: Exterior holomorphy and decay of the Cauchy transform
  • proof
  • Theorem 6: Rigidity
  • proof
  • ...and 5 more