Functions in R²(E) and points of the fine interior

Edwin Wolf

Vol. 134 (1988), No. 2, 393–400

DOI: 10.2140/pjm.1988.134.393

Abstract

Let E ⊂ ℂ be a set that is compact in the usual topology. Let m denote 2-dimensional Lebesgue measure. We denote by R₀(E) the algebra of rational functions with poles off E. For p ≥ 1, let L^p(E) = L^p(E,dm). The closure of R₀(E) in L^p(E) will be denoted by R^p(E).

In this paper we study the behavior of functions in R²(E) at points of the fine interior of E. We prove that if U ⊂ E is a finely open set of bounded point evaluations for R²(E), then there is a finely open set V ⊂ U such that each x ∈ V is a bounded point derivation of all orders for R²(E). We also prove that if R²(E)≠L²(E), there is a subset S ⊂ E having positive measure such that if x ∈ S each function in ⋃ _p>2R^p(E) is approximately continuous at x. Moreover, this approximate continuity is uniform on the unit ball of a normed linear space that contains ⋃ _p>2R^p(E).

Mathematical Subject Classification 2000

Primary: 41A20

Secondary: 30E10, 41A63, 46J10

Milestones

Received: 24 March 1987

Revised: 28 December 1987

Published: 1 October 1988