  Let $X$ be a compact subset of the complex plane and let $R^p(X)$, $2< p <
\infty$, denote the closure of the rational functions with poles off $X$ in the
$L^p$ norm. In this paper we consider three conditions that show how the
functions in $R^p(X)$ can have a greater degree of smoothness at the boundary
of $X$ than might otherwise be expected. We will show that two of the
conditions are equivalent and imply the third but the third does not imply the
other two.

Complex Variables;Functional Analysis