Aleman-Richter-Sundberg's Theorem On $P^t(μ)$-Spaces

Abstract

Let $ν$ be a finite complex measure with support in $\bar {\mathbb D}$ and let $\mathcal Cν$ denote the Cauchy transform of $ν.$ Suppose that $ν$ annihilates polynomials in complex variable $z$ and $ν|_{\partial \mathbb D} = hm,$ where $m$ is the normalized Lebesgue measure on $\partial {\mathbb D}$. We show that, for $ε_0 > 0,$ $m$-almost all $e^{iθ}\in \partial {\mathbb D},$ and $a > 0,$ when $r$ tends to 1, there exists $E_r \subset B(re^{iθ}, \frac{1-r}{4})$ with analytic capacity $γ(E_r) < ε_0 \frac{1-r}{4}$ such that $|\mathcal Cν(λ) - e^{-iθ}h(e^{iθ}) | \le a$ area-almost all $λ\in B (re^{iθ}, \frac{1-r}{4} ) \setminus E_r .$ Using this result, we provide an alternative proof of Aleman-Richter-Sundberg's Theorem on nontangential limits in $P^t(μ)$-Spaces and the index of invariant subspaces.