Nontangential limits in Pt(µ)-spaces and the index of invariant subgroups
(2009) In Annals of Mathematics 169(2). p.449-490- Abstract
- Abstract
Let μ be a finite positive
measure on the closed disk D¯
in the complex plane, let 1 ≤ t < ∞,
and let Pt(μ)
denote the closure of the analytic polynomials in
Lt(μ). We suppose
that D
is the set of analytic bounded point evaluations for
Pt(μ), and
that Pt(μ)
contains no nontrivial characteristic functions. It is then known that the restriction of
μ to
∂D must be of the form
h|dz|. We prove that every
function f ∈ Pt(μ) has nontangential
limits at h|dz|-almost
every point of ∂D,
and the... (More) - Abstract
Let μ be a finite positive
measure on the closed disk D¯
in the complex plane, let 1 ≤ t < ∞,
and let Pt(μ)
denote the closure of the analytic polynomials in
Lt(μ). We suppose
that D
is the set of analytic bounded point evaluations for
Pt(μ), and
that Pt(μ)
contains no nontrivial characteristic functions. It is then known that the restriction of
μ to
∂D must be of the form
h|dz|. We prove that every
function f ∈ Pt(μ) has nontangential
limits at h|dz|-almost
every point of ∂D,
and the resulting boundary function agrees with
f as an
element of Lt(h|dz|).
Our proof combines methods from James E. Thomson’s proof of the existence of bounded point
evaluations for Pt(μ)
whenever Pt(μ)≠Lt(μ)
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 ν
in the complex plane we are able to describe locations of bounded point evaluations
for Pt(ν) 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 < ∞ dim
ℳ∕zℳ = 1 for every nonzero
invariant subspace ℳ
of Pt(μ) if and
only if h≠0.
We also investigate the boundary behaviour of the functions in
Pt(μ) near the
points z ∈ ∂D
where h(z) = 0. In
particular, for 1 < t < ∞
we show that there are interpolating sequences for
Pt(μ)
that accumulate nontangentially almost everywhere on
{z : h(z) = 0}. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1467113
- author
- Aleman, Alexandru LU ; Richter, Stefan and Sundberg, Carl
- organization
- publishing date
- 2009
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Annals of Mathematics
- volume
- 169
- issue
- 2
- pages
- 449 - 490
- publisher
- Annals of Mathematics
- external identifiers
-
- scopus:71649104937
- ISSN
- 0003-486X
- language
- English
- LU publication?
- yes
- id
- bbe3a04b-1640-4e05-a55e-9333d23f70a8 (old id 1467113)
- date added to LUP
- 2016-04-01 12:12:28
- date last changed
- 2022-03-21 00:55:47
@article{bbe3a04b-1640-4e05-a55e-9333d23f70a8,
abstract = {{Abstract<br/><br>
<br/><br>
<br/><br>
<br/><br>
<br/><br>
<br/><br>
<br/><br>
<br/><br>
Let μ be a finite positive<br/><br>
measure on the closed disk D¯<br/><br>
in the complex plane, let 1 ≤ t < ∞,<br/><br>
and let Pt(μ)<br/><br>
denote the closure of the analytic polynomials in<br/><br>
Lt(μ). We suppose<br/><br>
that D<br/><br>
is the set of analytic bounded point evaluations for<br/><br>
Pt(μ), and<br/><br>
that Pt(μ)<br/><br>
contains no nontrivial characteristic functions. It is then known that the restriction of<br/><br>
μ to<br/><br>
∂D must be of the form<br/><br>
h|dz|. We prove that every<br/><br>
function f ∈ Pt(μ) has nontangential<br/><br>
limits at h|dz|-almost<br/><br>
every point of ∂D,<br/><br>
and the resulting boundary function agrees with<br/><br>
f as an<br/><br>
element of Lt(h|dz|).<br/><br>
<br/><br>
Our proof combines methods from James E. Thomson’s proof of the existence of bounded point<br/><br>
evaluations for Pt(μ)<br/><br>
whenever Pt(μ)≠Lt(μ)<br/><br>
with Xavier Tolsa’s remarkable recent results on analytic capacity. These methods allow<br/><br>
us to refine Thomson’s results somewhat. In fact, for a general compactly supported<br/><br>
measure ν<br/><br>
in the complex plane we are able to describe locations of bounded point evaluations<br/><br>
for Pt(ν) in<br/><br>
terms of the Cauchy transform of an annihilating measure.<br/><br>
<br/><br>
As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We<br/><br>
show that for 1 < t < ∞ dim<br/><br>
ℳ∕zℳ = 1 for every nonzero<br/><br>
invariant subspace ℳ<br/><br>
of Pt(μ) if and<br/><br>
only if h≠0.<br/><br>
<br/><br>
We also investigate the boundary behaviour of the functions in<br/><br>
Pt(μ) near the<br/><br>
points z ∈ ∂D<br/><br>
where h(z) = 0. In<br/><br>
particular, for 1 < t < ∞<br/><br>
we show that there are interpolating sequences for<br/><br>
Pt(μ)<br/><br>
that accumulate nontangentially almost everywhere on<br/><br>
{z : h(z) = 0}.}},
author = {{Aleman, Alexandru and Richter, Stefan and Sundberg, Carl}},
issn = {{0003-486X}},
language = {{eng}},
number = {{2}},
pages = {{449--490}},
publisher = {{Annals of Mathematics}},
series = {{Annals of Mathematics}},
title = {{Nontangential limits in Pt(µ)-spaces and the index of invariant subgroups}},
volume = {{169}},
year = {{2009}},
}