Boundary values in $R^t(K,\mu)$-spaces and invariant subspaces

TL;DR

This paper investigates boundary behaviors of functions in $R^t(K, u)$-spaces, extending previous results on nontangential limits and providing new insights into the continuity of Cauchy transforms of annihilating measures.

Contribution

It extends the work of Aleman, Richter, and Sundberg on boundary limits to $R^t(K, u)$-spaces and introduces new continuity properties of Cauchy transforms of annihilating measures.

Findings

Extended boundary value results for $R^t(K, u)$-spaces.

Showed continuity properties of Cauchy transforms in capacitary density sense.

Provided alternative proofs of existing theorems in the field.

Abstract

For $1 \le t < \infty ,$ a compact subset $K$ of the complex plane $\mathbb C,$ and a finite positive measure $\mu$ supported on $K,$ $R^t(K, \mu)$ denotes the closure in $L^t (\mu )$ of rational functions with poles off $K.$ The paper examines the boundary values of functions in $R^t(K, \mu)$ for certain compact subset $K$ and extends the work of Aleman, Richter, and Sundberg on nontangential limits for the closure in $L^t (\mu )$ of analytic polynomials (Theorem A and Theorem C in \cite{ars}). We show that the Cauchy transform of an annihilating measure has some continuity properties in the sense of capacitary density. This allows us to extend Aleman, Richter, and Sundberg's results for $R^t(K, \mu)$ and provide alternative short proofs of their theorems as special cases.