We define the Dirichlet
space 𝒟 on the unit polydisc 𝕌n of ℂn. 𝒟 is a semi-Hilbert space of holomorphic
functions, contains the holomorphic polynomials densely, is invariant under
compositions with the biholomorphic automorphisms of 𝕌n, and its semi-norm is
preserved under such compositions. We show that 𝒟 is unique with these properties.
We also prove 𝒟 is unique if we assume that the semi-norm of a function in 𝒟
composed with an automorphism is only equivalent in the metric sense to the
semi-norm of the original function. Members of a subclass of 𝒟 given by a norm can
be written as potentials of ℒ2-functions on the n-torus 𝕋n. We prove that the
functions in this subclass satisfy strong-type inequalities and have tangential limits
almost everywhere on ∂𝕌n. We also make capacitory estimates on the size of the
exceptional sets on ∂𝕌n.
