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Czechoslovak Mathematical Journal

, Volume 68, Issue 2, pp 559–576 | Cite as

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

  • Boban Karapetrović
Article
  • 47 Downloads

Abstract

We show that if α > 1, then the logarithmically weighted Bergman space \(A_{{{\log }^\alpha }}^2\) is mapped by the Libera operator L into the space \(A_{{{\log }^{\alpha - 1}}}^2\), while if α > 2 and 0 < εα−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 L maps the logarithmically weighted Bloch space \({B_{{{\log }^\alpha }}}\), α ∈ R, into itself, while H maps \({B_{{{\log }^\alpha }}}\) into \({B_{{{\log }^{\alpha + 1}}}}\).

In Pavlović’s paper (2016) it is shown that L maps the logarithmically weighted Hardy-Bloch space \(B_{{{\log }^\alpha }}^1\), α > 0, into \(B_{{{\log }^{\alpha - 1}}}^1\). We show that this result is sharp. We also show that H maps \(B_{{{\log }^\alpha }}^1\), α > 0, into \(B_{{{\log }^{\alpha - 1}}}^1\) and that this result is sharp also.

Keywords

Libera operator Hilbert matrix operator Hardy space Bergman space Bloch space Hardy-Bloch space 

MSC 2010

47B38 47G10 30H25 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of BelgradeBelgradeSerbia

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