  This paper is devoted to various applications of Hardy-Sobolev type
inequalities. We derive a new $L^2$ estimate for the $\bar{\partial}-$equation
on ${\mathbb C}^n$ which yields a quantitative generalization of the Hartogs
extension theorem to the case when the singularity set is not necessary
compact. We show that for any negative subharmonic function $\psi$ on ${\mathbb
R}^n$, $n>2$, the BMO norm of $\log |\psi|$ is bounded above by $2\sqrt{n-2}$
and $|\psi|^\gamma$ satisfies a reverse H\"older inequality for every
$0<\gamma<1$. We also show that every plurisubharmonic function is locally BMO.
Several Liouville theorems for subharmonic functions on complete Riemannian
manifolds are given. As a consequence, we get a Margulis type theorem that if a
bounded domain in ${\mathbb C}^n$ covers a Zariski open set in a projective
algebraic variety, then the group of deck transformations of the covering has
trivial center.

Complex Variables;Analysis of PDEs