Linus Bergqvist: Holomorphic functions on polydiscs and measures on the distinguished boundary

Abstract.

This thesis consists of four papers, treating holomorphic functions defined on polydiscs, operators acting on spaces of such functions, and related measures on the \(n\)-torus.

In paper I we study containment of rational inner functions, or RIFs for short, in Dirichlet type spaces on the unit polydisc \(\mathbb{D}^n\). In particular, a theorem relating \(H^p\) integrability of the partial derivatives of an RIF to containment of the function in certain Dirichlet type spaces is proved. As a corollary we see that every RIF on \(\mathbb{D}^n\) belongs to the isotropic Dirichlet type space with parameter \(1/n\). We also show that if \(\phi = \tilde{p}/p\) is an RIF on \(\mathbb{D}^n\) with the property that \(1/p\) lies in some isotropic Dirichlet type space with parameter \(\alpha < 0\), then \(\phi\) is contained in the isotropic Dirichlet type space with parameter \(\alpha+2/n\).

In paper II we provide new proofs of Mandrekar's theorem on shift invariant subspaces of the Hardy space \(H^2(\mathbb{D}^2)\). The theorem says that an invariant subspace \(\mathcal{M}\) of \(H^2(\mathbb{D}^2)\) is generated by an inner function if and only if the shift operators are doubly commuting on \(\mathcal{M}\). The new proofs in this paper are elementary and transparent, and mainly use basic properties of reproducing kernels.

In paper III we study Clark measures corresponding to RIFs on \(\mathbb{D}^2\). We give an explicit description of such measures when regarded as bounded linear functionals on the continuous functions on \(\mathbb{T}^2\), analyse when the corresponding Clark embedding operator is unitary, and relate the density of these measures to a geometric property of zero sets associated with the corresponding RIFs.

In paper IV we study measures on \(\mathbb{T}^n\) having the property that their Poisson integral is the real part of some holomorphic function on \(\mathbb{T}^n\): so called RP-measures. We give necessary conditions on the support of RP-measures, and among other things show that their supports cannot have linear measure zero. Furthermore, we relate failure of a set to support any positive RP-measure with uniform approximability of continuous functions by certain holomorphic functions. For \(n=2\) this gives us a necessary and sufficient condition for a subset of \(\mathbb{T}^2\) to contain the support of some positive RP-measure.

Read the thesis on DiVA