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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2602.13734 |
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| _version_ | 1866915798474293248 |
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| author | Kalaj, David |
| author_facet | Kalaj, David |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13734 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $L^2$-Norm of the Cauchy transform on circular annuli Kalaj, David Complex Variables 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. |
| title | The $L^2$-Norm of the Cauchy transform on circular annuli |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2602.13734 |