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Hilbert type operators acting from the Bloch space into Bergman spaces
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 68 / Issue 1 / February 2025
- Published online by Cambridge University Press:
- 26 November 2024, pp. 205-225
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- Article
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Let µ be a finite positive Borelmeasure on
$[0,1)$ and 
$\alpha \gt -1$. The generalized integral operator of Hilbert type 
$\mathcal {I}_{\mu_{\alpha+1}}$ is defined on the spaces 
$H(\mathbb{D})$ of analytic functions in the unit disc 
$\mathbb{D}$ as follows:
\begin{equation*}\mathcal {I}_{\mu_{\alpha+1}}(f)(z)=\int_{0}^{1} \frac{f(t)}{(1-tz)^{\alpha+1}}d\mu(t),\ \ f\in H(\mathbb{D}),\ \ z\in \mathbb{D} .\end{equation*}In this paper, we give a unified characterization of the measures µ for which the operator
$\mathcal {I}_{\mu_{\alpha+1}}$ is bounded from the Bloch space to a Bergman space for all 
$\alpha \gt -1$. Additionally, we also investigate the action of 
$\mathcal {I}_{\mu_{\alpha+1}}$ from the Bloch space to the Hardy spaces and the Besov spaces.
