FINELY HOLOMORPHIC-FUNCTIONS

Journal article

Let O(U) denote the finely harmonic functions on U a finely open subset of C such that ∂g ∂ z ̄ = 0 almost surely on U. Define Af(K) to be those g in C(K) such that if K′ is the fine interior of K then g |K′ is in O(K). We prove that Af(K) is invariant under the Vitushkin localization operators, i.e., it is T-invariant. We also settle an open question of Fuglede on the existence of polygonal arcs in finely open subsets of Rn. Using T-invariance we prove that point derivation yields a continuous functional on Af(K) for each point of K′. Using the polygonal arc results as well one can show that for a large class of z in K′, f(ω) = (g(ω) - g(z)) (ω - z) extends across z and is in O(K), from which f(z) = g′(z). We also establish that R(K) and Af(K) have the same Arens-Singer and Jensen measures. © 1980.

Authors: Lyons, T

• Publication status: Published
• Journal: JOURNAL OF FUNCTIONAL ANALYSIS
• Volume: 37
• Issue: 1
• Pages: 1-18
• Publication date: 1980-01-01
• DOI: 10.1016/0022-1236(80)90024-5
• EISSN: 1096-0783
• ISSN: 0022-1236
• Language: English