Bounded point derivations on $R^p(X)$ and approximate derivatives

TL;DR

The paper proves that bounded point derivations on certain function spaces imply the existence of approximate derivatives at specific points, extending previous results from uniform to $L^p$ norms for $p > 2$.

Contribution

It extends Wang's result from uniform algebras to $L^p$ spaces, establishing a link between bounded point derivations and approximate derivatives for $p > 2$.

Findings

Bounded point derivations imply approximate derivatives at the same point.

Extension of Wang's result from uniform to $L^p$ spaces for $p > 2$.

Higher order bounded point derivations also imply approximate derivatives.

Abstract

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 $p >2$, then there is an approximate derivative at $x_0$. A similar result is proven for higher order bounded point derivations. This extends a result of Wang which was proven for $R(X)$, the uniform closure of rational functions with poles off $X$.