A note on composition operators on the bidisc

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 $\mathrm{\Phi }(z_1,z_2)=({\phi }_1(z_1),{\phi }_2(z_2))$. We also consider the problem of boundedness of composition operators $C_{\mathrm{\Phi }}:\mathfrak{D}({\mathbb{D}}^2)\to A^2({\mathbb{D}}^2)$ for general self maps of the bidisc, applying some recent results about Carleson measures on the the Dirichlet space of the bidisc.