Operators of Hilbert type acting on some spaces of analytic functions

Let $H(\mathbb{D})$ be the space of all analytic functions in the unit disc $\mathbb{D}$. For $g\in H(\mathbb{D})$, the generalized Hilbert operator ${\mathcal{H}}_g$ is defined by $${\mathcal{H}}_g(f)(z)={\int }_0^{1}f(t)g^{\mathrm{\prime }}(tz)dt,z\in \mathbb{D},f\in H(\mathbb{D})\mathrm{.}$$ In this paper, we study the operator ${\mathcal{H}}_g$ acting on some spaces of analytic functions in $\mathbb{D}$. Specifically, we give a complete characterization of those $g\in H(\mathbb{D})$ for which the operator ${\mathcal{H}}_g$ is bounded (resp. compact) from the Dirichlet space ${\mathcal{D}}_{\alpha }^2$ to ${\mathcal{D}}_{\beta }^2$ for all possible indicators $\alpha ,\beta \in \mathbb{R}$. We also study the action of the operator ${\mathcal{H}}_g$ on the space of bounded analytic functions $H^{\mathrm{\infty }}$, which generalizes the known results for the classical Hilbert operator $\mathcal{H}$ acting on $H^{\mathrm{\infty }}$. In particular, we consider the boundedness of the operator ${\mathcal{H}}_g$ with a symbol of non-negative Taylor coefficients, acting on logarithmic Bloch spaces and on Korenblum spaces. This work generalizes the corresponding results for the classical Hilbert operator.