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

### Abstract

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 $L^t(h|dz|)$.

Our proof combines methods from James E. Thomson’s proof of the existence of bounded point evaluations for $P^t(\mu)$ whenever $P^t(\mu) \neq L^t(\mu)$ with Xavier Tolsa’s remarkable recent results on analytic capacity. These methods allow us to refine Thomson’s results somewhat. In fact, for a general compactly supported measure $\nu$ in the complex plane we are able to describe locations of bounded point evaluations for $P^t(\nu)$ in terms of the Cauchy transform of an annihilating measure.

As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We show that for $1 < t < \infty$ dim $\mathcal M/z\mathcal M = 1$ for every nonzero invariant subspace $\mathcal M$ of $P^t(\mu)$ if and only if $h \ne 0$.

We also investigate the boundary behaviour of the functions in $P^t(\mu)$ near the points $z \in \partial \mathbb D$ where $h(z) = 0$. In particular, for $1 < t < \infty$ we show that there are interpolating sequences for $P^t(\mu)$ that accumulate nontangentially almost everywhere on $\{z:h(z)=0\}$.

## Authors

Alexandru Aleman

Center for Mathematical Sciences
Lund University
22100 Lund
Sweden

Stefan Richter

Department of Mathematics
University of Tennessee
Knoxville, TN 37996
United States

Carl Sundberg

Department of Mathematics
University of Tennessee
Knoxville, TN 37996
United States