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 Hölder 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.

A $\partial \bar{\partial }-$proof of the Hartogs extension theorem for pluriharmonic functions is added

Cited by in corpus (2)

• $L^2$ estimates of Poincaré-Lelong equations on convex domains in ${\mathbb{C}}^n$
• On the Poincare-Lelong equation in ${\mathbb{C}}^n$