On Lattice Structure of the Space of Pointwise Limits of Harmonic Functions

Abstract

Let H 1 (U ) denote the space of all pointwise limits of bounded sequences from H(U ), where H(U ) consists of all continuous functions on the closure U of a bounded open set U ⊂ R m that are harmonic on U . It is shown that the space H 1 (U ) is a lattice in the natural ordering if and only if the set ∂ reg U of all regular points of U is an F σ -set. Mathematics Subject Classifications (2000): 31B05, 54C30.