Libera and Hilbert matrix operator on logarithmically weighted Bergman, Bloch and Hardy-Bloch spaces

@article{Karapetrović2018,
abstract = {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 $0<\varepsilon \le \alpha -2$, then the Hilbert matrix operator $H$ maps $A_\{\log ^\alpha \}^2$ into $A_\{\log ^\{\alpha -2-\varepsilon \}\}^2$.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 $\mathcal \{B\}_\{\log ^\{\alpha +1\}\}$.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 $\mathcal \{B\}_\{\log ^\{\alpha \}\}^1$, $\alpha \ge \{0\}$, into $\mathcal \{B\}_\{\log ^\{\alpha -1\}\}^1$ and that this result is sharp also.},
author = {Karapetrović, Boban},
journal = {Czechoslovak Mathematical Journal},
keywords = {Libera operator; Hilbert matrix operator; Hardy space; Bergman space; Bloch space; Hardy-Bloch space},
language = {eng},
number = {2},
pages = {559-576},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Libera and Hilbert matrix operator on logarithmically weighted Bergman, Bloch and Hardy-Bloch spaces},
url = {http://eudml.org/doc/294342},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Karapetrović, Boban
TI - Libera and Hilbert matrix operator on logarithmically weighted Bergman, Bloch and Hardy-Bloch spaces
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 2
SP - 559
EP - 576
AB - 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 $0<\varepsilon \le \alpha -2$, then the Hilbert matrix operator $H$ maps $A_{\log ^\alpha }^2$ into $A_{\log ^{\alpha -2-\varepsilon }}^2$.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 $\mathcal {B}_{\log ^{\alpha +1}}$.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 $\mathcal {B}_{\log ^{\alpha }}^1$, $\alpha \ge {0}$, into $\mathcal {B}_{\log ^{\alpha -1}}^1$ and that this result is sharp also.
LA - eng
KW - Libera operator; Hilbert matrix operator; Hardy space; Bergman space; Bloch space; Hardy-Bloch space
UR - http://eudml.org/doc/294342
ER -