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Libera and Hilbert matrix operator on logarithmically weighted Bergman, Bloch and Hardy-Bloch spaces
Czechoslovak Mathematical Journal, Tome 68 (2018) no. 2, pp. 559-576 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We show that if $\alpha >1$, then the logarithmically weighted Bergman space $A_{\log ^{\alpha }}^2$ is mapped by the Libera operator $\mathcal {L}$ into the space $A_{\log ^{\alpha -1}}^2$, while if $\alpha >2$ and $0\varepsilon \leq \alpha -2$, then the Hilbert matrix operator $H$ maps $A_{\log ^\alpha }^2$ into $A_{\log ^{\alpha -2-\varepsilon }}^2$.\newline We show that the Libera operator $\mathcal {L}$ maps the logarithmically weighted Bloch space $\mathcal {B}_{\log ^{\alpha }}$, $\alpha \in \mathbb {R}$, into itself, while $H$ maps $\mathcal {B}_{\log ^{\alpha }}$ into $\mathcal {B}_{\log ^{\alpha +1}}$.\newline In Pavlović's paper (2016) it is shown that $\mathcal {L}$ maps the logarithmically weighted Hardy-Bloch space $\mathcal {B}_{\log ^{\alpha }}^1$, $\alpha >0$, into $\mathcal {B}_{\log ^{\alpha -1}}^1$. We show that this result is sharp. We also show that $H$ maps $\mathcal {B}_{\log ^{\alpha }}^1$, $\alpha \geq {0}$, into $\mathcal {B}_{\log ^{\alpha -1}}^1$ and that this result is sharp also.
We show that if $\alpha >1$, then the logarithmically weighted Bergman space $A_{\log ^{\alpha }}^2$ is mapped by the Libera operator $\mathcal {L}$ into the space $A_{\log ^{\alpha -1}}^2$, while if $\alpha >2$ and $0\varepsilon \leq \alpha -2$, then the Hilbert matrix operator $H$ maps $A_{\log ^\alpha }^2$ into $A_{\log ^{\alpha -2-\varepsilon }}^2$.\newline We show that the Libera operator $\mathcal {L}$ maps the logarithmically weighted Bloch space $\mathcal {B}_{\log ^{\alpha }}$, $\alpha \in \mathbb {R}$, into itself, while $H$ maps $\mathcal {B}_{\log ^{\alpha }}$ into $\mathcal {B}_{\log ^{\alpha +1}}$.\newline In Pavlović's paper (2016) it is shown that $\mathcal {L}$ maps the logarithmically weighted Hardy-Bloch space $\mathcal {B}_{\log ^{\alpha }}^1$, $\alpha >0$, into $\mathcal {B}_{\log ^{\alpha -1}}^1$. We show that this result is sharp. We also show that $H$ maps $\mathcal {B}_{\log ^{\alpha }}^1$, $\alpha \geq {0}$, into $\mathcal {B}_{\log ^{\alpha -1}}^1$ and that this result is sharp also.
DOI : 10.21136/CMJ.2018.0555-16
Classification : 30H25, 47B38, 47G10
Keywords: Libera operator; Hilbert matrix operator; Hardy space; Bergman space; Bloch space; Hardy-Bloch space 
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     title = {Libera and {Hilbert} matrix operator on logarithmically weighted {Bergman,} {Bloch} and {Hardy-Bloch} spaces},
     journal = {Czechoslovak Mathematical Journal},
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     year = {2018},
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Karapetrović, Boban. Libera and Hilbert matrix operator on logarithmically weighted Bergman, Bloch and Hardy-Bloch spaces. Czechoslovak Mathematical Journal, Tome 68 (2018) no. 2, pp. 559-576. doi: 10.21136/CMJ.2018.0555-16

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