Mathematical Sciences

Lund University

Title Nontangential limits in Pt(µ)-spaces and the index of invariant subgroups
Authors Alexandru Aleman, Stefan Richter, Carl Sundberg
Publication Annals of Mathematics
Year 2009
Volume 169
Issue 2
Pages 449 - 490
Document type Article
Status Published
Quality controlled Yes
Language eng
Publisher Princeton University Press
Abstract English Abstract<br> <br> <br> <br> <br> <br> <br> <br> Let μ be a finite positive<br> measure on the closed disk D¯<br> in the complex plane, let 1 ≤ t &lt; ∞,<br> and let Pt(μ)<br> denote the closure of the analytic polynomials in<br> Lt(μ). We suppose<br> that D<br> is the set of analytic bounded point evaluations for<br> Pt(μ), and<br> that Pt(μ)<br> contains no nontrivial characteristic functions. It is then known that the restriction of<br> μ to<br> ∂D must be of the form<br> h|dz|. We prove that every<br> function f ∈ Pt(μ) has nontangential<br> limits at h|dz|-almost<br> every point of ∂D,<br> and the resulting boundary function agrees with<br> f as an<br> element of Lt(h|dz|).<br> <br> Our proof combines methods from James E. Thomson’s proof of the existence of bounded point<br> evaluations for Pt(μ)<br> whenever Pt(μ)≠Lt(μ)<br> with Xavier Tolsa’s remarkable recent results on analytic capacity. These methods allow<br> us to refine Thomson’s results somewhat. In fact, for a general compactly supported<br> measure ν<br> in the complex plane we are able to describe locations of bounded point evaluations<br> for Pt(ν) in<br> terms of the Cauchy transform of an annihilating measure.<br> <br> As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We<br> show that for 1 &lt; t &lt; ∞ dim<br> ℳ∕zℳ = 1 for every nonzero<br> invariant subspace ℳ<br> of Pt(μ) if and<br> only if h≠0.<br> <br> We also investigate the boundary behaviour of the functions in<br> Pt(μ) near the<br> points z ∈ ∂D<br> where h(z) = 0. In<br> particular, for 1 &lt; t &lt; ∞<br> we show that there are interpolating sequences for<br> Pt(μ)<br> that accumulate nontangentially almost everywhere on<br> {z : h(z) = 0}.
ISBN/ISSN/Other ISSN: 0003-486X print