A note on composition operators on the bidisc
Athanasios Beslikas
TL;DR
This work investigates when composition operators act boundedly on Dirichlet-type spaces on the disc and bidisc, and between the Dirichlet and Bergman spaces on the bidisc. It introduces a two-dimensional change-of-variables technique and leverages the generalized Nevanlinna counting function to derive new boundedness criteria, including a coordinate-wise characterization for separated symbols on the bidisc. A key contribution is a Carleson-measure framework that yields a one-box sufficient condition for C_{\Phi}:\mathfrak{D}(\mathbb{D}^2)\to A^2_{\beta}(\mathbb{D}^2) and a comprehensive necessary-and-sufficient criterion via capacity for general holomorphic maps, connecting operator boundedness to Carleson measures on bidisc Dirichlet spaces. The results establish a bridge between Carleson-measure theory, capacity on the bidisc, and composition-operator boundedness, and they open avenues for extending to anisotropic spaces and other product domains. The practical impact lies in providing concrete, checkable conditions to determine boundedness of composition operators in several complex variables models relevant to function theory and operator theory.
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 $Φ(z_1,z_2)=(φ_1(z_1),φ_2(z_2))$. We also consider the problem of boundedness of composition operators $C_Φ:\mathfrak{D}(\mathbb{D}^2)\to A^2(\mathbb{D}^2)$ for general self maps of the bidisc, applying some recent results about Carleson measures on the the Dirichlet space of the bidisc.