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