Optimal $L^p$ Hardy-type inequalities

Abstract

Let $\Omega$ be a domain in ${\mathbb{R}}^n$ or a noncompact Riemannian manifold of dimension $n\ge 2$, and $1<p<\infty$. Consider the functional $\mathcal{Q}(\phi ):={\int }_{\Omega }(|\nabla \phi {|}^p+V|\phi {|}^p)\mathrm{d}\nu$ defined on $C_0^{\infty }(\Omega )$, and assume that $\mathcal{Q}\ge 0$. The aim of the paper is to generalize to the quasilinear case ($p\ne 2$) some of the results obtained in [6] for the linear case ($p=2$), and in particular, to obtain “as large as possible” nonnegative (optimal) Hardy-type weight $W$ satisfying

Our main results deal with the case where $V=0$, and $\Omega$ is a general punctured domain (for $V\ne 0$ we obtain only some partial results). In the case $1<p\le n$, an optimal Hardy-weight is given by

where $G$ is the associated positive minimal Green function with a pole at $0$. On the other hand, for $p>n$, several cases should be considered, depending on the behavior of $G$ at infinity in $\Omega$. The results are extended to annular and exterior domains.