The Cauchy transform on bounded domains

Anderson, J. M.; Hinkkanen, A.

doi:10.1090/S0002-9939-1989-0972226-5

The Cauchy transform on bounded domains
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by J. M. Anderson and A. Hinkkanen

Proc. Amer. Math. Soc. 107 (1989), 179-185

DOI: https://doi.org/10.1090/S0002-9939-1989-0972226-5

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Abstract:

Suppose that $f$ is in ${L^2}(\Delta )$ where $\Delta$ is the unit disk, and that $f = 0$ outside $\Delta$. We show that then the Cauchy transform $\mathcal {C} f$ of $f$, when restricted to $\Delta$, satisfies $||\mathcal {C} f|{|_2} \leq (2/\alpha )||f|{|_2}$, where $\alpha \approx 2.4048$ is the smallest positive zero of the Bessel function ${J_0}$. This inequality is sharp.

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