Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sharp Hardy inequalities involving distance functions from submanifolds of Riemannian manifolds

Ningwei Cui, Alexandru Kristály, Wei Zhao

TL;DR

The paper develops sharp Barbatis–Filippas–Tertikas–type Hardy inequalities on complete Riemannian manifolds by linking singular weights to both submanifold geometry and ambient curvature through distance functions. Central to the approach are HK-type Laplacian comparison estimates for the distance to a submanifold and a weak-divergence framework, enabling curved Hardy inequalities with double-curvature weights and, in a logarithmic form, with sharp constants. The authors establish sharpness and nonexistence of extremals, extend results to larger function spaces, and provide two principal applications: Hardy inequalities under upper curvature bounds with double-curvature and log-weight variants, and analogous inequalities under lower curvature bounds. The results unify and extend classical Euclidean Hardy inequalities, recovering them in suitable limits and offering new curvature-aware tools for analysis on manifolds with submanifolds.

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.

Sharp Hardy inequalities involving distance functions from submanifolds of Riemannian manifolds

TL;DR

The paper develops sharp Barbatis–Filippas–Tertikas–type Hardy inequalities on complete Riemannian manifolds by linking singular weights to both submanifold geometry and ambient curvature through distance functions. Central to the approach are HK-type Laplacian comparison estimates for the distance to a submanifold and a weak-divergence framework, enabling curved Hardy inequalities with double-curvature weights and, in a logarithmic form, with sharp constants. The authors establish sharpness and nonexistence of extremals, extend results to larger function spaces, and provide two principal applications: Hardy inequalities under upper curvature bounds with double-curvature and log-weight variants, and analogous inequalities under lower curvature bounds. The results unify and extend classical Euclidean Hardy inequalities, recovering them in suitable limits and offering new curvature-aware tools for analysis on manifolds with submanifolds.

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.
Paper Structure (20 sections, 47 theorems, 233 equations)

This paper contains 20 sections, 47 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Submanifold geometry & Fermi coordinates
  4. Heintze--Karcher-type Laplacian estimates
  5. Weak divergence
  6. A general Hardy inequality
  7. Abstract construction
  8. Sharpness
  9. Non-existence of extremal functions
  10. Extensions to larger function spaces
  11. Extension to the space $C^\infty_0(\Omega)$
  12. Extension to the space $C^\infty_0(\Omega,\Sigma)$
  13. Application I: Sharp Hardy inequalities on manifolds with upper curvature bounds
  14. Hardy inequalities involving double-curvature weights
  15. Hardy inequalities involving logarithmic double-curvature weights
  16. ...and 5 more sections

Key Result

Proposition 2.1

Given $m,n\in \mathbb{N}$ and $\lambda,\kappa\in \mathbb{R}$, let Define a function $G_{\lambda,\kappa}:(0,\mathfrak{t}_{\lambda,\kappa})\rightarrow \mathbb{R}$ by If $m\geq n+1$, then $G_{\lambda,\kappa}$ is decreasing in both $\lambda$ and $\kappa$, i.e., for any $\lambda\leq \Lambda$ and $\kappa\leq K$,

Theorems & Definitions (111)

  • Proposition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Lemma 2.3: Jacobi Criteria
  • Lemma 2.4
  • Remark 2.1
  • Example 2.1
  • ...and 101 more