Approximation in the mean by rational functions II (1907.04287v3)

Abstract: For $1\le t < \infty$, a compact subset $K\subset\mathbb C$, 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 $K$. Conway and Yang (2019) introduced the concept of non-removable boundary $\mathcal F$ and removable set $\mathcal R = K\setminus \mathcal F$ for $R^t(K, \mu)$. We continue the previous work and obtain structural results for $R^t(K, \mu)$. Assume that $S_\mu$, the multiplication by $z$ on $R^t(K, \mu)$, is pure ($R^t(K, \mu)$ does not have $L^t$ summand). Let $H^\infty_{\mathcal R}(A_{\mathcal R})$ be the weak$^*$ closure in $L^\infty (A_{\mathcal R})$ of the functions that are bounded analytic off compact subsets of $\mathcal F$, where $A_{\mathcal R}$ denotes the area measure restricted to $\mathcal R$. $\mathcal R$ is $\gamma$-connected ($\gamma$ denotes analytic capacity) if for any two disjoint open set $G_1$ and $G_2$ with $\mathcal R \subset G_1 \cup G_2 ~\gamma-a.a.$, then $\mathcal R \subset G_1 ~\gamma-a.a.$ or $\mathcal R \subset G_2 ~\gamma-a.a.$. We prove: (1) $R^t(K, \mu)$ contains no non-trivial characterization functions if and only if the removable set $\mathcal R$ is $\gamma$-connected. (2) There is an isometric isomorphism and a weak$^*$ homeomorphism from $R^t(K, \mu)\cap L^\infty(\mu )$ onto $H^\infty_{\mathcal R}(A_{\mathcal R })$.