Möbius-invariant Hilbert spaces in polydiscs (Q1322107)

We define the Dirichlet space \({\mathcal D}\) on the unit polydisc \(U^ n\) of \(\mathbb{C}^ n\). \({\mathcal D}\) is a semi-Hilbert space of holomorphic functions, contains the holomorphic polynomials densely, is invariant under compositions with the holomorphic automorphisms of \(U^ n\), and its seminorm is preserved under such compositions. We show that \({\mathcal D}\) is unique with these properties. We also prove \({\mathcal D}\) is unique if we assume that the seminorm of a function in \(D\) composed with an automorphism is only equivalent in the metric sense to the seminorm of the original function. Members of a subclass of \({\mathcal D}\) given by a norm can be written as potentials of \(L^ 2\)-functions on the \(n\)-torus \(T^ n\). We prove that the functions in this subclass satisfy strong- type inequalities and have tangential limits almost everywhere on \(\partial U^ n\). We also make capacitory estimates on the size of the exponential sets on \(\partial U^ n\).