Sharp multipolar $L^p$-Hardy-type inequalities on Riemannian manifolds
Cristian Ciulică, Teodor Rugină
TL;DR
The paper develops sharp multipolar $L^p$-Hardy-type inequalities on complete Riemannian manifolds for $1<p<N$ by employing a supersolution approach in conjunction with curvature comparison theorems. It introduces a multipolar potential $V_{p,a_1,...,a_n}$ built from distance functions to the poles and proves a general lower bound $\int_M |\nabla_g u|^p dv_g \ge \int_M V_{p,a_1,...,a_n} |u|^p dv_g$, with an explicit bipolar reduction that yields positivity on Cartan-Hadamard manifolds and a sharp constant in the corresponding minimization problem. The authors further derive curvature-enhanced remainders in the Cartan-Hadamard setting and examine constant-curvature cases, including hyperbolic space and the upper hemisphere, where the remainders depend explicitly on curvature and geometry. These results extend Hardy-type inequalities beyond the Euclidean setting, illuminate curvature effects on multipolar singular potentials, and have potential implications for $p$-Laplacian PDEs on manifolds.
Abstract
In this paper we prove sharp multipolar Hardy-type inequalities in the Riemannian $L^p-$setting for $p\geq 2$ using the method of super-solutions and fundamental results from comparison theory on manifolds, thus generalizing previous results for $p=2$. We emphasize that when we restrict to Cartan-Hadamard manifolds, the inequalities improve in the case $2<p<N$ compared to the case $p=2$ since we obtain positive remainder terms which are controlled by curvature estimates. In the end, we treat the cases of positive and negative constant sectional curvature.