Superposition operators, Hardy spaces, and Dirichlet type spaces

Abstract

For $0<p<\infty$ and $\alpha >-1$ the space of Dirichlet type ${\mathcal{D}}_{\alpha }^p$ 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 ${\mathcal{D}}_{\alpha }^p$. 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 $\phi$ such that the superposition operator $S_{\phi }$ with symbol $\phi$ 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 ${\mathcal{D}}_{\alpha }^p$ into $Q_s$ and $Q_s$ into ${\mathcal{D}}_{\alpha }^p$, for any admissible triplet of numbers $(p,\alpha ,s)$.

Cited by in corpus (1)

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