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Keywords:
Libera operator; Hilbert matrix operator; Hardy space; Bergman space; Bloch space; Hardy-Bloch space
Libera operator; Hilbert matrix operator; Hardy space; Bergman space; Bloch space; Hardy-Bloch space
Summary:
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.
References:
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