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 $L^t(h|dz|)$.
Alexandru Aleman 1 ; Stefan Richter 2 ; Carl Sundberg 2
@article{10_4007_annals_2009_169_449,
author = {Alexandru Aleman and Stefan Richter and Carl Sundberg},
title = {Nontangential limits in $\mathcal{P}^t(\mu)$-spaces and the index of invariant subgroups},
journal = {Annals of mathematics},
pages = {449--490},
year = {2009},
volume = {169},
number = {2},
doi = {10.4007/annals.2009.169.449},
mrnumber = {2480609},
zbl = {1179.46020},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.169.449/}
}
TY - JOUR
AU - Alexandru Aleman
AU - Stefan Richter
AU - Carl Sundberg
TI - Nontangential limits in $\mathcal{P}^t(\mu)$-spaces and the index of invariant subgroups
JO - Annals of mathematics
PY - 2009
SP - 449
EP - 490
VL - 169
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UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.169.449/
DO - 10.4007/annals.2009.169.449
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%J Annals of mathematics
%D 2009
%P 449-490
%V 169
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.169.449/
%R 10.4007/annals.2009.169.449
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Alexandru Aleman; Stefan Richter; Carl Sundberg. Nontangential limits in $\mathcal{P}^t(\mu)$-spaces and the index of invariant subgroups. Annals of mathematics, Tome 169 (2009) no. 2, pp. 449-490. doi: 10.4007/annals.2009.169.449
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