Weighted Hardy inequality on Riemannian manifolds

TL;DR

This paper investigates the existence of minimizers for weighted Hardy inequalities on compact Riemannian manifolds with singular weights along submanifolds, providing necessary and sufficient conditions for their existence.

Contribution

It establishes new criteria for the existence and non-existence of minimizers of Hardy inequalities with singular weights on Riemannian manifolds.

Findings

Necessary and sufficient conditions for minimizer existence.

Characterization of singular weight functions on submanifolds.

Extension of Hardy inequality theory to Riemannian manifolds.

Abstract

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $N\geq 3$ and we let $\Sigma$ to be a closed submanifold of dimension $1 \leq k \leq N-2. $ In this paper we study existence and non-existence of minimizers of Hardy inequality with weight function singular on $\Sigma$ within the framework of Brezis-Marcus-Shafrir. In particular we provide necessary and sufficient conditions for existence of minimizers.