Superharmonic functions and bounded point evaluations (Q582419)

Summary: Let E be a compact subset of the complex plane \({\mathbb{C}}\). We denote by \(R_ 0(E)\) the algebra consisting of the (restrictions to E of) rational functions with poles off E. Let m denote the 2-dimensional Lebesgue measure. Let \(R^ 2(E)\) be the closure of \(R_ 0(E)\) in \(L^ 2(E,dm).\) We consider points \(x\in E\) such that ``evaluation at x'' extends from \(R_ 0(E)\) to a continuous linear functional on \(R^ 2(E)\). These points are bounded point evaluations on \(R^ 2(E)\). Hedberg, Fernström and Polking used capacity to identity bounded point evaluations. We use their results to show that the existence of a bounded point evaluation \(x\in E\) is equivalent to the existence of a superharmonic function u(y) that grows sufficiently fast as y approaches x through the complement of E.