English

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 < e ≤ 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 Pavlovic’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.