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Generalized integration operator between the Bloch-type space and weighted Dirichlet-type spaces. (English) Zbl 07345749

Let \(H(\mathbb D)\) be the set of holomorphic functions on the \(\mathbb D\), the unit complex disc, \(\alpha>0\), the \(\alpha\)-Bloch space \((\mathcal{B}^\alpha,\|\cdot\|_{\mathcal{B}^\alpha})\) be the space of functions \(f\in H(\mathbb D)\) such that \(\|f\|_{\mathcal{B}^\alpha}= \displaystyle\sup_{z\in\mathbb D}[(1-|z|^2)^\alpha|f'(z)|]<\infty\). Let \((p,\beta)\in (0,\infty)\times (-1,\infty)\) and \((\mathcal{D}^p_\beta,\|\cdot\|_{\mathcal{D}_\beta^p})\) be the weighted Dirichlet-type space comprised by functions \(f\in H(\mathbb D)\) such that \(\|f\|_{\mathcal{D}_\beta^p}^p=\displaystyle\int_{\mathbb D}|f'(z)|^p\left(\log(|z|^{-1}\right)^\beta dA(z)<\infty\) where \(dA(z)\) represents the normalized Lebesgue area measure on the unit complex disc. The purpose of the authors is to study the boundedness and the compactness of \(I_{g,\varphi}^{(n)}\), the \(\mathcal{D}_\beta^p\) (resp. \(\mathcal{B}^\alpha\))-valued operator on \(\mathcal{B}^\alpha\) (resp. \(\mathcal{D}_\beta^p\)) defined as \(I_{g,\varphi}^{(n)}(f)(z)=\displaystyle\int_0^zf^{(n)}(\varphi(\xi))g(\xi)d\xi\) where \(\varphi\) and \(g\) are a holomorphic self-map of \(\mathbb D\) and a holomorphic function on \(\mathbb D\), respectively.

MSC:

47G10 Integral operators
30H30 Bloch spaces
30H99 Spaces and algebras of analytic functions of one complex variable
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