Riccati Pairs ̶ New Approach for Hardy–Rellich Inequalities

This monograph presents alternative approaches to Hardy and Rellich inequalities. Concerning Hardy-type inequalities on Riemannian manifolds, we establish a generic functional inequality, both in additive and multiplicative forms, which produces well- known and genuinely new inequalities. For the additive version we introduce the notion of Riccati pairs, which enables us to give short/elegant proofs for several celebrated functional inequalities on Riemannian manifolds with sectional curvature bounded from above, by simply solving a Riccati-type ODE. The multiplicative version allows us to cover uncertainty principles as well.