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<p>We show that the range of the Cauchy transform on the classical Bergman space is the reproducing kernel Hilbert space of functions in the bi-analytic Dirichlet space subject to a first-order differential equation. The closed formula of its reproducing kernel is given explicitly. Our investigation includes an explicit characterization of the range of the weighted Cauchy transform on the weighted Bergman–Dirichlet spaces, extending Dyn’kin’s result showing that the single-valued functions in the Dirichlet space vanishing at the origin are the Cauchy transform of some functions in the Bergman space of anti-holomorphic functions.</p>

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The Cauchy Transform on the Bergman–Dirichlet Spaces

  • 期刊论文

  • 发表日期：

    2025年6月3日

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  • Abdelhadi Benahmadi,
  • Abdellatif Elkachkouri,
  • Allal Ghanmi

摘要

We show that the range of the Cauchy transform on the classical Bergman space is the reproducing kernel Hilbert space of functions in the bi-analytic Dirichlet space subject to a first-order differential equation. The closed formula of its reproducing kernel is given explicitly. Our investigation includes an explicit characterization of the range of the weighted Cauchy transform on the weighted Bergman–Dirichlet spaces, extending Dyn’kin’s result showing that the single-valued functions in the Dirichlet space vanishing at the origin are the Cauchy transform of some functions in the Bergman space of anti-holomorphic functions.