ON LIOUVILLE THEOREMS FOR HARMONIC FUNCTIONS WITH FINITE DIRICHLET INTEGRAL
Alexander Asaturovich Grigor'yan · Mathematics of the USSR-Sbornik · 1988
A criterion for the validity of the -Liouville theorem is proved. In §1 it is shown that the question of - and -Liouville theorems reduces to the study of the so-called massive sets (in other words, the level sets of harmonic functions in the classes and ). In §2 some properties of capacity are presented. In §3 the criterion of -massiveness is formulated — the central result of this article — and examples are presented. In §4 a criterion for the -Liouville theorem is formulated, and corollaries are derived. In §§5-9 the main theorems are proved.Figures: 5. Bibliography: 17 titles.