Optimal L ^p Hardy-type inequalities

Devyver, Baptiste; Pinchover, Yehuda

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 1, pp. 93-118.

Let Ω be a domain in ${\mathbb{R}}^n$ or a noncompact Riemannian manifold of dimension $n\ge 2$, and $1<p<\infty$. Consider the functional $\mathcal{Q}(\phi ):={\int }_{\Omega }({|\mathrm{\nabla }\phi |}^p+V{|\phi |}^p)\mathrm{d}\nu$ 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

$$\mathcal{Q}(\phi )\ge \underset{\Omega }{\int }W{|\phi |}^p\mathrm{d}\nu \forall \phi \in C_0^{\infty }(\Omega ).$$

Our main results deal with the case where $V=0$, and Ω 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

$$W:={\left(\frac{p-1}{p}\right)}^p{\left|\frac{\mathrm{\nabla }G}{G}\right|}^p,$$

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.

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