Generalized Hilbert Operators

If $g$ is an analytic function in the unit disc $\D$ we consider the generalized Hilbert operator $\hg$ defined by {equation*}\label{H-g} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt. {equation*} We study these operators acting on classical spaces of analytic functions in $\D$. More precisely, we address the question of characterizing the functions $g$ for which the operator $\hg$ is bounded (compact) on the Hardy spaces $H^p$, on the weighted Bergman spaces $A^p_\alpha$ or on the spaces of Dirichlet type $\mathcal D^p_\alpha$.