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

Time: Fri 2023-09-15 13.00

Location: Albano campus, lärosal 32, house 4

Doctoral student: Linus Bergqvist , Department of Mathematics, Stockholm University

Opponent: Emmanuel Fricain (Université de Lille)

Supervisor: Alan Sola

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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