Nontangential limits in Pt(µ)-spaces and the index of invariant subgroups
Author
Summary, in English
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}.
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}.
Department/s
Publishing year
2009
Language
English
Pages
449-490
Publication/Series
Annals of Mathematics
Volume
169
Issue
2
Document type
Journal article
Publisher
Annals of Mathematics
Topic
- Mathematics
Status
Published
ISBN/ISSN/Other
- ISSN: 0003-486X