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

David Kalaj

Abstract

We compute the exact $L^2$ operator norm of the Cauchy transform \[ (C_Ωf)(z)=\frac1π\int_Ω\frac{f(w)}{z-w}\,dA(w) \] on a circular annulus $Ω=A(r,R)=\{r<|z|<R\}$. Exploiting rotational symmetry and a Fourier mode decomposition, we reduce the problem to a one--dimensional weighted Hardy operator and obtain \[ \|C_{A(r,R)}\|_{L^2\to L^2} = \frac{2}{\sqrt{μ_1^{ND}(r,R)}}, \] where $μ_1^{ND}(r,R)$ 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.

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

Abstract

We compute the exact operator norm of the Cauchy transform on a circular annulus . Exploiting rotational symmetry and a Fourier mode decomposition, we reduce the problem to a one--dimensional weighted Hardy operator and obtain where is the first eigenvalue of the Laplacian on with Neumann condition on the inner boundary and Dirichlet condition on the outer boundary. The extremizers are explicitly described in terms of Bessel functions.
Paper Structure (10 sections, 8 theorems, 62 equations, 2 figures)

This paper contains 10 sections, 8 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and related work
  3. The proof of main results
  4. Bessel functions and the characteristic equation.
  5. Dirichlet--Cauchy operator and its sharp $L^2$ norm
  6. Competing interests.
  7. Data availability.
  8. Ethical approval.
  9. Consent to participate.
  10. Consent for publication.

Key Result

Proposition 1

Let $\Omega\subset\mathbb{R}^2$ be a smoothly bounded domain. Then where $\lambda_1(\Omega)$ is the smallest positive eigenvalue for the boundary value problem (1.2) in $\Omega$.

Figures (2)

  • Figure 1: $2/\kappa_{1,1}(r)$ for $R=1$, $0\le r<1$.
  • Figure 2: Plot of the normalized quantity $2/k_1(r)$ for $R=1$ and $0<r<1$.

Theorems & Definitions (16)

  • Proposition 1: Norm estimate for the Cauchy transform
  • Theorem 2: Main result
  • Lemma 3: Exact action on a single Fourier mode
  • proof
  • Lemma 4: Weighted Hardy reduction for $m\ge 1$
  • proof
  • Lemma 5: Bessel form for $m\ge 1$
  • proof
  • Lemma 6
  • proof
  • ...and 6 more