Sharp Hardy inequalities via Riemannian submanifolds

TL;DR

This paper establishes sharp weighted Hardy inequalities related to distance functions from submanifolds in Riemannian manifolds, extending previous Euclidean and Riemannian results and identifying optimal function spaces.

Contribution

It introduces new sharp Hardy inequalities for submanifolds of arbitrary codimension in Riemannian manifolds with non-negative curvature, including compact cases.

Findings

Sharp weighted Hardy inequalities are proven for various submanifold cases.

Optimal function spaces for Hardy inequalities in compact manifolds are identified.

The results extend and complement existing inequalities in Euclidean and Riemannian geometries.

Abstract

This paper is devoted to Hardy inequalities concerning distance functions from submanifolds of arbitrary codimensions in the Riemannian setting. On a Riemannian manifold with non-negative curvature, we establish several sharp weighted Hardy inequalities in the cases when the submanifold is compact as well as non-compact. In particular, these inequalities remain valid even if the ambient manifold is compact, in which case we find an optimal space of smooth functions to study Hardy inequalities. Further examples are also provided. Our results complement in several aspects those obtained recently in the Euclidean and Riemannian settings.