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Superharmonic functions and bounded point evaluations. (English) Zbl 0691.31001
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.
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
46E15 Banach spaces of continuous, differentiable or analytic functions
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