# Optimal $L_{p}$ Hardy-type inequalities

### Baptiste Devyver

Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel### Yehuda Pinchover

Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel

## Abstract

Let $Ω$ be a domain in $R_{n}$ or a noncompact Riemannian manifold of dimension $n≥2$, and $1<p<∞$. Consider the functional $Q(φ):=∫_{Ω}(∣∇φ∣_{p}+V∣φ∣_{p})dν$ defined on $C_{0}(Ω)$, and assume that $Q≥0$. The aim of the paper is to generalize to the quasilinear case ($p=2$) 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 $Ω$ is a general punctured domain (for $V=0$ we obtain only some partial results). In the case $1<p≤n$, 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 $Ω$. The results are extended to annular and exterior domains.

## Cite this article

Baptiste Devyver, Yehuda Pinchover, Optimal $L_{p}$ Hardy-type inequalities. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 1, pp. 93–118

DOI 10.1016/J.ANIHPC.2014.08.005