Superposition operators, Hardy spaces, and Dirichlet type spaces

For $0<p<\infty $ and $\alpha >-1$ the space of Dirichlet type $\mathcal D^p_\alpha $ consists of those functions $f$ which are analytic in the unit disc $\mathbb D$ and satisfy $\int_{\mathbb D}(1-| z| )^\alpha| f^\prime (z)| ^p\,dA(z)<\infty $. The space $\Dp$ is the closest one to the Hardy space $H^p$ among all the $\mathcal D^p_\alpha $. Our main object in this paper is studying similarities and differences between the spaces $H^p$ and $\Dp$ ($0<p<\infty $) regarding superposition operators. Namely, for $0<p<\infty $ and $0<s<\infty $, we characterize the entire functions $\varphi $ such that the superposition operator $S_\varphi $ with symbol $\varphi $ maps the conformally invariant space $Q_s$ into the space $\Dp$, and, also, those which map $\Dp$ into $Q_s$ and we compare these results with the corresponding ones with $H^p$ in the place of $\Dp$. We also study the more general question of characterizing the superposition operators mapping $\mathcal D^p_\alpha $ into $Q_s$ and $Q_s$ into $\mathcal D^p_\alpha $, for any admissible triplet of numbers $(p, \alpha , s)$.