Möbius-invariant Hilbert spaces in polydiscs

Abstract

We define the Dirichlet space 31 on the unit polydisc of . 2 is a semi-Hilbert space of of holomorphic functions, contains the holomorphic polynomials densely, is invariant under compositions with the biholomorphic automorphisms of , and its semi-norm is preserved under such compositions. We show that 2 is unique with these properties. We also prove 3ί is unique if we assume that the semi-norm of a function in 3f composed with an automorphism is only equivalent in the metric sense to the semi-norm of the original function. Members of a subclass of 3f given by a norm can be written as potentials of «5 5>2-functions on the «-torus . We prove that the functions in this subclass satisfy strong-type inequalities and have tangential limits almost everywhere on dVn . We also make capacitory estimates on the size of the exceptional sets on f)Vn . 1. Introduction. Mόbius-invariant spaces. Let U be the open unit disc in C and T be the unit circle bounding it. The open unit polydisc Vn and the torus Tn in Cn are the cartesian products of n unit discs and n unit circles, respectively. Ίn is the distinguished boundary of Uw and forms only a small part of the topological boundary dVn of Un . We denote by Jΐ the group of all biholomorphic automorphisms of Vn (the Mδbius group). The subgroup of linear automorphisms in Jf is denoted by %. The space of holomorphic functions with domain Un will be called ^(ϋn) and will carry the topology of uniform convergence on compact subsets of Vn. A semi-inner product on a complex vector space β? is a sesquilinear functional on β(? x %? with all the properties of an inner product except that it is possible to have {(a, a)) = 0 when a Φ 0. IWI = \/(( α> a)) *s the associated semi-norm. We assume ((• , •)) is