Bounded point derivations on R^p(X) and approximate derivatives - Deterding, Stephen (Mathematica Scandinavica, 2019)
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.