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In this note we characterise the bounded composition operators on the anisotropic Dirichlet-type spaces D₀ (D²) induced by holomorphic self maps of the bi-disc D² of the form (z₁, z₂) = (₁ (z₁), ₂ (z₂) ) via the respective generalised Nevanlinna counting functions of both symbols. We also characterise the bounded composition operators from A² (D²) to D (D²) for general self maps of the bidisc via a Carleson-type condition.
Athanasios Beslikas (Thu,) studied this question.