On Liouville theorems for harmonic functions with finite Dirichlet integral

A. A. Grigor'yan

Sbornik. Mathematics, Tome 60 (1988) no. 2, pp. 485-504

Résumé

A criterion for the validity of the $D$-Liouville theorem is proved. In § 1 it is shown that the question of $L^\infty$ and $D$-Liouville theorems reduces to the study of the so-called massive sets (in other words, the level sets of harmonic functions in the classes $L^\infty$ and $L^\infty\cap D$). In § 2 some properties of capacity are presented. In § 3 the criterion of $D$-massiveness is formulated – the central result of this article – and examples are presented. In § 4 a criterion for the $D$-Liouville theorem is formulated, and corollaries are derived. §§ 5–9 the main theorems are proved. Figures: 5. Bibliography: 17 titles.