J. Rákosník (ed.),
Function spaces, differential operators and nonlinear analysis.
Proceedings of the conference held in Paseky na Jizerou, September 3-9, 1995.
Mathematical Institute, Czech Academy of Sciences, and Prometheus Publishing House, Praha 1996
p. 227 - 232

Compact operators and normal families

Andriy A. Kondratyuk, Svyatoslav I. Tarasyuk

Artema st. 59/14, 290053 Lviv, Ukraine kond@franko.lviv.ua Shevchenka st. 26/29 Dublyany, 292040 Zovkivskii, Lviv reg., Ukraine kond@franko.lviv.ua

Abstract: Properties of compact integral operators of potential type are applied to the theory of subharmonic functions.

Let $\overline{K}_R$ be a closed ball of radius $R$, $S^{m-1}$ the unit sphere in ${\Bbb R}^m$, $m\ge3$, $\{\mu_n\}$ a sequence of Borel measure in $K_R$, $$ P_n(x)= \int_{\overline{K}_R} \frac{d\mu_n(y)}{|x-y|^{m-2}}. $$

\proclaim{Theorem 1} Let a sequence of measures $\{\mu_n\}$ be weakly convergent to a measure $\mu$ in $\overline{K}_R$, then $$ \lim_{n\to \infty} \int_{S^{m-1}}|P_n(rx)-P(rx)|^qdS(x) = 0 $$ holds when $0<r<\infty$, $0\le q < 1 + \dfrac 1{m-2}$. \endproclaim

\proclaim{Theorem 2} Let ${\Cal F}$ be a family of subharmonic in ${\Bbb R}^m$ functions such that $$ \sup_{u\in {\Cal F}} \int_{S^{m-1}} |u(rx)|dS(x) < \infty $$ for every $r>0$. Then for every sequence $\{u_n\}$ from ${\Cal F}$ there exist a subharmonic in ${\Bbb R}^m$ function $v$ and a subsequence $\{u_{n_j}\}$ such that for every $r\in (0,\infty )$ and every $q$, $1\le q < 1+\dfrac 1{m-2}$, we have $$ \lim_{j\to\infty} \int_{S^{m-1}}|u_{n_j}(rx) - v(rx)|^qdS(x) = 0. $$ \endproclaim

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