A note on composition operators on the bidisc

TL;DR

This paper provides new criteria for the boundedness of composition operators on Dirichlet-type spaces on the bidisc, characterizing them for certain self maps and exploring boundedness conditions using Carleson measures.

Contribution

It introduces a new sufficient condition for boundedness via a two-dimensional change of variables and characterizes bounded composition operators induced by specific holomorphic self maps.

Findings

New sufficient condition for boundedness of composition operators.

Characterization of bounded operators induced by product-type self maps.

Application of Carleson measure results to boundedness problems.

Abstract

In this note we give a new sufficient condition for the boundedness of the composition operator on the Dirichlet-type space on the disc, via a two dimensional change of variables formula. With the same formula, we characterise the bounded composition operators on the anisotropic Dirichlet-type spaces $\mathfrak{D}_{\vec{a}}(\mathbb{D}^2)$ induced by holomorphic self maps of the bidisc $\mathbb{D}^2$ of the form $\Phi(z_1,z_2)=(\phi_1(z_1),\phi_2(z_2))$. We also consider the problem of boundedness of composition operators $C_{\Phi}:\mathfrak{D}(\mathbb{D}^2)\to A^2(\mathbb{D}^2)$ for general self maps of the bidisc, applying some recent results about Carleson measures on the the Dirichlet space of the bidisc.