Sharp Quantitative Forms of the Hardy Inequality on Cartan-Hadamard Manifolds via Sobolev-Lorentz Embeddings

TL;DR

The paper delivers a sharp quantitative refinement of the Hardy inequality on Cartan–Hadamard model manifolds by integrating Schwarz symmetrization with a volume-preserving Jacobian transformation. It introduces Lorentz-type spaces tailored to the manifold's volume growth, proves an optimal Sobolev–Lorentz embedding with an explicit best constant that is not attained, and provides a stability result: the Hardy deficit controls the distance to the family of virtual extremals in a Lorentz-type norm. A core innovation is linking manifold and Euclidean inequalities through a weighted Euclidean Hardy framework derived from a precise Jacobian change of variables, revealing how curvature dictates extremal behavior. These results extend Euclidean stability phenomena to curved spaces and offer a robust analytic toolkit for geometric Hardy-type inequalities anchored in model-manifold geometry.

Abstract

In this article, we investigate the quantitative form of the classical Hardy inequality. In our first result, we prove the following quantitative bound under the assumption that the $\mathbb{M}^N$ is a Riemannian model satisfying the centered isoperimetric inequality: We prove that $$ \|\nabla_g u\|^2_{L^{2}(\mathbb{M}^N)} - \frac{(N-2)^2}{4}\left\|\frac{u}{r(x)}\right\|^2_{L^2(\mathbb{M}^N)} \geq C [\mbox{dist}(u, Z)]^{\frac{4N}{N-2}}\left\|\frac{u}{r(x)}\right\|^2_{L^2(\mathbb{M}^N)},$$ for every real-valued weakly differentiable function $u$ on $\mathbb{M}^N$ such that $|\nabla_g u| \in L^2(\mathbb{M}^N)$ and $u$ decays to zero at infinity. Here $r(x) = d_g(x,x_0)$ denotes the geodesic distance from a fixed pole $x_0,$ the set $Z$ represents the family of virtual extremals, and the distance is understood in an appropriate generalized Lorentz-type space. Our approach is built on the symmetrization technique on manifolds, combined with a novel Jacobian-type transformation that provides a precise way for comparing volume growth, level sets, and gradient terms across the two geometries of Euclidean and manifold settings. When coupled with symmetrization, this framework yields sharp control over the relevant functionals and reveals how the underlying curvature influences extremal behavior. Our result generalizes the seminal result of Cianchi-Ferone [Ann. Inst. H. Poincaré C Anal. Non Linéaire 25 (2008)] to the curved spaces. Moreover, building upon this transformation, we succeed in extending Sobolev-Lorentz embedding-classically formulated in the Euclidean setting to the broader framework of Cartan-Hadamard models and we establish an optimal Sobolev-Lorentz embedding in this geometric setting. Finally, we establish a quantitative correspondence between the Hardy deficit on the manifold and an appropriate weighted Hardy deficit in Euclidean space, showing that each controls the other.

Theorems & Definitions (75)

• Theorem A: Cianchi and Ferone cf-aihp
• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Theorem 1.3
• Remark 1.3
• Theorem 1.4
• Remark 1.4
• Lemma 2.1