Superposition operators, Hardy spaces, and Dirichlet type spaces

For $0<p<\mathrm{\infty }$ and $\alpha >-1$ the space of Dirichlet type $\mathcal D^p_\alpha $consistsofthosefunctions$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 $\mathrm{.}Thespace$\Dp$istheclosestonetotheHardyspace$H^p$amongallthe$\mathcal D^p_\alpha $. Our main object in this paper is studying similarities and differences between the spaces $H^p$ and $\text{Dp}$ ($0<p<\mathrm{\infty }$) regarding superposition operators. Namely, for $0<p<\mathrm{\infty }$ and $0<s<\mathrm{\infty }$, we characterize the entire functions $\phi$ such that the superposition operator $S_{\phi }$ with symbol $\phi$ maps the conformally invariant space $Q_s$ into the space $\text{Dp}$, and, also, those which map $\text{Dp}$ into $Q_s$ and we compare these results with the corresponding ones with $H^p$ in the place of $\text{Dp}$. We also study the more general question of characterizing the superposition operators mapping ${\mathcal{D}}_{\alpha }^p$ into $Q_s$ and $Q_s$ into ${\mathcal{D}}_{\alpha }^p$, for any admissible triplet of numbers $(p, \alpha , s)$.