Hardy-Sobolev type inequalities and their applications

This paper is devoted to various applications of Hardy-Sobolev type inequalities. We derive a new $L^2$ estimate for the $\bar{\mathrm{\partial }}-$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 ${\mathbb R}^n$,$n>2$,theBMOnormof$\log |\psi|$isboundedaboveby$2\sqrt{n-2}$and$|\psi|^\gamma$satisfiesareverseH\ddot{o}lderinequalityforevery$0<\gamma<1$. 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.