A formula for a bounded point derivation on $R^p(X)$

TL;DR

This paper extends a theorem on bounded point derivations for rational function closures, showing they can be represented by difference quotients under certain conditions in $L^p$ spaces for $p > 2$.

Contribution

It generalizes O'Farrell's theorem to $L^p$ spaces, providing a formula for bounded point derivations on $R^p(X)$ when $X$ has an interior cone.

Findings

Bounded point derivations can be represented by difference quotients.

Results hold for $p > 2$ and sets with an interior cone.

Extension of O'Farrell's theorem to $L^p$ spaces.

Abstract

Let $X$ be a compact subset of the complex plane. It is shown that if a point $x_0$ admits a bounded point derivation on $R^p(X)$, the closure of rational function with poles off $X$ in the $L^p(dA)$ norm, for $p >2$ and if $X$ contains an interior cone, then the bounded point derivation can be represented by the difference quotient if the limit is taken over a non-tangential ray to $x_0$. A similar result is proven for higher order bounded point derivations. These results extend a theorem of O'Farrell for $R(X)$, the closure of rational functions with poles off $X$ in the uniform norm.