# Linus Lidman Bergqvist: Function spaces and rational inner functions on polydiscs

Time: Thu 2022-01-13 09.00 - 11.00

Location: Kräftriket, House 6, Room 306 and Zoom: 682 9011 8600 (password needed)

Doctoral student: Linus Lidman Bergqvist

Opponent: Jan-Fredrik Olsen (Lund)

Supervisor: Alan Sola

### Abstract

In this thesis we consider problems related to rational inner functions and several different Hilbert spaces on the unit polydisc. In the general introduction the functions and the function spaces we will be interested in are introduced, and in particular we point out problems and phenomena that occur in higher dimensions and are not present for one variable functions. For example, we provide a detailed construction of a non-trivial shift-invariant subspace of Dirichlet-type spaces on the bidisc which is not finitely generated. Furthermore, Clark–Aleksandrov measures are generalized to higher dimensions, and certain results about such measures are proved.

Paper I concerns containment of rational inner functions in Dirichlet-type spaces on polydiscs. In particular a theorem relating $$H^p$$ integrability of the partial derivatives of a rational inner function to containment of the function in certain Dirichlet-type spaces is proved. As a corollary, we see that every rational inner function on $$D^n$$ belongs to the isotropic Dirichlet-type space with weight $$\frac{1}{n}$$. In Paper II, Zhu's sub-Bergman spaces of one variable functions on the unit disc are generalized to weighted Bergman spaces on $$D^n$$. Unlike in one variable, we show that sub-Bergman spaces associated to a rational inner function are generally not contained in a weighted Bergman space of higher regularity. We also show how Clark measures on the n-torus can be used to study model spaces on $$D^n$$ associated to rational inner functions.