Functional inequalities of hardy-rellich type in euclidean and non-euclidean settings

In this PhD thesis we study functional inequalities associated with degenerate elliptic and subelliptic operators, with a particular focus on Hardy, Rellich, and Hardy-Rellich inequalities. Our analysis covers both Euclidean and non-Euclidean settings, including operators such as the Kohn Laplacian on the Heisenberg group and the Grushin operator. The work develops a unified approach based on integral identities, which extends classical results to the \(L^p\) setting and to weighted or degenerate contexts. Within this general theory, we establish sufficient conditions for the validity of Hardy, Rellich, and Hardy-Rellich type inequalities with general weights, and identify the sharp constants whenever possible. Applications include weighted and unweighted inequalities of logarithmic, cylindrical, and Gaussian type, Brezis-Vázquez-type improvements, explicit lower bounds for the optimal constant in the unweighted Hardy inequality on the Heisenberg group, and explicit formulas and improvements for domains with \(0 \in \partial \Omega\) in Grushin-type structures.