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 $\{z:h(z)=0\}$.
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