Superposition operators, Hardy spaces, and Dirichlet type spaces

TL;DR

This paper investigates the behavior of superposition operators between Dirichlet type spaces and conformally invariant spaces, providing characterizations and comparisons with Hardy spaces for various parameter ranges.

Contribution

It offers new characterizations of superposition operators mapping between Dirichlet type spaces and $Q_s$ spaces, extending known results for Hardy spaces.

Findings

Characterization of entire functions for superposition operators from $Q_s$ to $ ext{Dirichlet}$ spaces.

Comparison of superposition operator behavior between $H^p$ and $ ext{Dirichlet}$ spaces.

General results for operators between $ ext{Dirichlet}$ spaces and $Q_s$ spaces for various parameters.

Abstract

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 $(p, \alpha , s)$.