Improved Hardy inequalities on Riemannian Manifolds

We study the following version of Hardy-type inequality on a domain $\Omega$ in a Riemannian manifold $(M,g)$: $$\int{\Omega}|\nabla u|_g^p\rho^\alpha dV_g \geq \left(\frac{|p-1+\beta|}{p}\right)^p\int{\Omega}\frac{|u|^p|\nabla \rho|_g^p}{|\rho|^p}\rho^\alpha dV_g +\int{\Omega} V|u|^p\rho^\alpha dV_g, \quad \forall\ u\in C_c^\infty (\Omega).$$ We provide sufficient conditions on $p, \alpha, \beta,\rho$ and $V$ for which the above inequality holds. This generalizes earlier well-known works on Hardy inequalities on Riemannian manifolds. The functional setup covers a wide variety of particular cases, which are discussed briefly: for example, $\mathbb{R}^N$ with $p<N$, $\mathbb{R}^N\setminus \{0\}$ with $p\geq N$, $\mathbb{H}^N$, etc.