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}(