Abstract
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 Qian and Li 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 \(C_{\mathcal {B}_{\log }}(\mathcal {D}_\alpha \cap \mathcal {B}_{\log })\).
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Communicated by H. Turgay Kaptanoglu.
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The work was supported by NNSF of China (Nos. 11801347, 12071272 and 11720101003), NSF of Guangdong Province (No. 2018A030313512), Key projects of fundamental research in universities of Guangdong Province (No. 2018KZDXM034) and Shantou University SRFT (No. NTF17020)
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Bao, G., Lou, Z. & Zhou, X. Closure in the Logarithmic Bloch Norm of Dirichlet Type Spaces. Complex Anal. Oper. Theory 15, 74 (2021). https://doi.org/10.1007/s11785-021-01121-7
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DOI: https://doi.org/10.1007/s11785-021-01121-7