Sharp Hardy inequalities involving distance functions from submanifolds of Riemannian manifolds

TL;DR

This paper develops new Hardy inequalities on Riemannian manifolds involving submanifold distance functions, utilizing curvature bounds and comparison techniques, and establishes sharpness and extremal function properties.

Contribution

It introduces novel Hardy inequalities in curved spaces using geometric analysis and comparison methods, extending classical inequalities to Riemannian manifolds with curvature considerations.

Findings

Derived Hardy inequalities with curvature-dependent weights

Proved sharpness and non-existence of extremal functions under mild conditions

Unified classical Hardy inequalities as special cases

Abstract

We establish various Hardy inequalities involving the distance function from submanifolds of Riemannian manifolds, where the natural weights are expressed in terms of bounds of the mean curvature of the submanifold and sectional/Ricci curvature of the ambient Riemannian manifold. Our approach is based on subtle Heintze-Karcher-type Laplace comparisons of the distance function and on a D'Ambrosio-Dipierro-type weak divergence formula for suitable vector fields, providing Barbatis-Filippas-Tertikas-type Hardy inequalities in the curved setting. Under very mild assumptions, we also establish the sharpness and non-existence of extremal functions within the Hardy inequalities and - depending on the geometry of the ambient manifold - their extensibility to various function spaces. Several examples are provided by showing the applicability of our approach; in particular, well-known Hardy inequalities appear as limit cases of our new inequalities.