Abstract

Let E be a compact subset of the complex plane ℂ. We denote by R(E) the algebra consisting of (the restrictions to E of) rational functions with poles off E. Let m denote 2-dimensional Lebesgue measure. For p≥1, let Rp(E) be the closure of R(E) in Lp(E,dm).In this paper we consider the case p=2. Let x ϵ ∂E be a bounded point evaluation for R2(E). Suppose there is a C>0 such that x is a limit point of the set s={y|y ϵ Int E,Dist(y,∂E)≥C|y−x|}. For those y ϵ S sufficiently near x we prove statements about |f(y)−f(x)| for all f ϵ R(E).