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Closure in the logarithmic Bloch norm of Dirichlet type spaces. (English) Zbl 1482.46028

Summary: In this paper, for every \(\alpha \in \mathbb{R} \), we characterize \(C_{\mathcal{B}_{\log }}(\mathcal{D}_\alpha \cap \mathcal{B}_{\log })\), the closure of Dirichlet type space \(\mathcal{D}_\alpha\) in the logarithmic Bloch space \(\mathcal{B}_{\log } \). For the case of \(\alpha =0\), we answer a question raised by R.-S. Qian and S.-X. Li [Indag. Math., New Ser. 29, No. 5, 1432–1440 (2018; Zbl 1489.47045)] recently. We also consider the strict inclusion relation among the little logarithmic Bloch space, \(C_{\mathcal{B}_{\log }}(\mathcal{D}_\alpha \cap \mathcal{B}_{\log })\) and \(\mathcal{B}_{\log } \). In addition, we revisit a description of the boundedness of composition operator from \(\mathcal{B}_{\log }\) to \(C_{\mathcal{B}_{\log }}(\mathcal{D}_\alpha \cap \mathcal{B}_{\log })\).

MSC:

46E15 Banach spaces of continuous, differentiable or analytic functions
30H30 Bloch spaces
30J10 Blaschke products
47B33 Linear composition operators

Citations:

Zbl 1489.47045
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Full Text: DOI

References:

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