## 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 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}.

Original language | English |
---|---|

Pages (from-to) | 449-490 |

Journal | Annals of Mathematics |

Volume | 169 |

Issue number | 2 |

Publication status | Published - 2009 |

## Subject classification (UKÄ)

- Mathematics