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 Alogα2 is mapped by the Libera operator L into the space Alogα−12, while if α > 2 and 0 < ε ≤ α−2, then the Hilbert matrix operator H maps Alogα2 into Alogα−2−ε2. We show that the Libera operator L maps the logarithmically weighted Bloch space Blogα, α ∈ R, into itself, while H maps Blogα into Blogα+1. In Pavlović’s paper (2016) it is shown that L maps the logarithmically weighted Hardy-Bloch space Blogα1, α > 0, into Blogα−11. We show that this result is sharp. We also show that H maps Blogα1, α > 0, into Blogα−11 and that this result is sharp also.