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

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 < ε ≤ α−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.