Hardy-Sobolev type inequalities and their applications

Abstract

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.