Various types of convergence of sequences of $\delta$-subharmonic functions

A. F. Grishin ; A. Chouigui

Sbornik. Mathematics, Tome 199 (2008) no. 6, pp. 811-832

Résumé

Let $v_n(z)$ be a sequence of $\delta$-subharmonic functions in some domain $G$. Conditions are studied under which the convergence of $v_n(z)$ as a sequence of generalized functions implies its convergence in the Lebesgue spaces $L_p(\gamma)$. Hörmander studied the case where $v_n(z)$ is a sequence of subharmonic functions and the measure $\gamma$ is the restriction of the Lebesgue measure to a compactum contained in $G$. In this paper a more general case is considered and theorems of two types are obtained. In theorems of the first type it is assumed that $\operatorname{supp}\gamma\Subset G$. In theorems of the second type it is assumed that the support of the measure is a compactum and $\operatorname{supp}\gamma\subset\overline G$. In the second case, $G$ is assumed to be the half-plane. Bibliography: 11 titles.