A class of symbols that induce bounded composition operators for Dirichlet-type spaces on the disc

TL;DR

This paper studies when holomorphic self-maps $\varphi$ of the unit disc induce bounded composition operators on Dirichlet-type spaces $\mathcal{D}_p(\mathbb{D})$. It introduces a bidisc-based approach using the lift operator $L^{p,\gamma}$ and a De Branges–Rovnyak kernel $k^{\varphi}$, deriving a sufficient condition: if $\|k^{\varphi}\|_{\infty}<\infty$, then $C_{\varphi}$ is bounded on $\mathcal{D}_{2\sigma-2\beta}(\mathbb{D})$ for $p=2\sigma-2\beta$, under $\sigma>0$ and $\tfrac{\sigma}{2}-1<\beta<\sigma$. The argument combines a boundedness result for $C_{\Phi}$ on $A_{\sigma}^2(\mathbb{D}^2)$ with Balooch’s double-integral characterization to control the one-dimensional norm, effectively transferring multidimensional estimates to the disc. This highlights the De Branges–Rovnyak kernel as a natural criterion and links Dirichlet-type space theory with bidisc Bergman-space techniques.

Abstract

In this note we study the problem of determining the holomorphic self maps of the unit disc that induce a bounded composition operator on Dirichlet-type spaces. We find a class of symbols $\varphi$ that induce a bounded composition operator on the Dirichlet-type spaces, by applying results of the multidimensional theory of composition operators for the weighted Bergman spaces of the bi-disc.

Theorems & Definitions (10)

• Theorem 2.1
• Definition 1
• Proposition 1
• proof
• Theorem 2.2
• Lemma 2.1
• proof
• Theorem 3.1
• proof
• Remark 1