Nontangential limits in $\mathcal{P}^t(\mu)$-spaces and the index of invariant subgroups

Alexandru Aleman¹ ; Stefan Richter² ; Carl Sundberg²

¹ Center for Mathematical Sciences<br/>Lund University<br/>22100 Lund<br/>Sweden
² Department of Mathematics<br/>University of Tennessee<br/>Knoxville, TN 37996<br/>United States

Annals of mathematics, Tome 169 (2009) no. 2, pp. 449-490.

Résumé

Let $\mu$ be a finite positive measure on the closed disk $\overline{\mathbb D}$ in the complex plane, let $1 \le t < \infty$, and let $P^t(\mu)$ denote the closure of the analytic polynomials in $L^t(\mu)$. We suppose that $\mathbb D$ is the set of analytic bounded point evaluations for $P^t(\mu)$, and that $P^t(\mu)$ contains no nontrivial characteristic functions. It is then known that the restriction of $\mu$ to $\partial \mathbb D$ must be of the form $h|dz|$. We prove that every function $f \in P^t(\mu)$ has nontangential limits at $h|dz|$-almost every point of $\partial \mathbb D$, and the resulting boundary function agrees with $f$ as an element of $L^t(h|dz|)$.

Détail

DOI : 10.4007/annals.2009.169.449