Bounded point derivations on certain function spaces

Let X be a compact nowhere dense subset of the complex plane C, and let dA denote two-dimensional Lebesgue (or area) measure in C. Denote by R(X) the set of all rational functions having no poles on X, and by R^p(X) the closure of R(X) in Lp(X, dA) whenever 1. p < ∞. The purpose of this paper is to study the relationship between bounded derivations on R^p(X) and the existence of approximate derivatives provided 2 < p < ∞ and to draw attention to an anomaly that occurs when p = 2.