Hardy and Rellich identities and inequalities for Baouendi-Grushin operators via spherical vector fields

For Baouendi-Grushin vector fields, we prove Hardy, Hardy-Rellich, and Rellich identities and inequalities with sharp constants. Our explicit remainder terms significantly improve than those found in the literature. Our arguments are built on abstract Hardy-Rellich identities involving the Bessel pair along with the use of spherical harmonics developed by Garofalo-Shen [Ann. Inst. Fourier (1994)]. Furthermore, in the spirit of Bez-Machihara-Ozawa [Math. Z (2023)], we construct spherical vector fields corresponding to the Baouendi-Grushin vector fields and prove identities that, in turn, establish optimal Rellich identities, by comparing the Baouendi-Grushin operator with its radial and spherical components. We give alternate proofs of Hardy identities and inequalities with enhanced Hardy constants in some subspaces of the Sobolev space, among other things. Additionally, we compute the deficit involving the $L^2$-norm of the Baouendi-Grushin operator and its radial component with an explicit remainder term, which leads to a comparison of the Baouendi-Grushin operator with its radial components. As a consequence of the main results, new second-order Heisenberg-Pauli-Weyl uncertainty principles and Hydrogen uncertainty principles are also derived. Furthermore, we also derive certain symmetrization principles green corresponding to the Baouendi-Grushin vector fields.