Improved Hardy inequalities on Riemannian manifolds

We study the following version of Hardy-type inequality on a domain Ω in a Riemannian manifold (Formula presented.) : (Formula presented.) We provide sufficient conditions on (Formula presented.) 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, (Formula presented.) with p<N, (Formula presented.) with (Formula presented.), (Formula presented.), etc.