Hardy and Rellich Inequalities with Bessel Pairs

In this paper, we establish suitable characterisations for a pair of functions $(W(x),H(x))$ on a bounded, connected domain Ω ⊂ Rⁿ in order to have the following Hardy inequality: {equation*} ∫Ω W(x) |∇ u|_A^2 dx ≥ ∫Ω |∇ d|^2_AH(x)|u|^2 dx, \,\,\, u ∈ C¹_0(Ω), {equation*} where d ( x ) is a suitable quasi-norm (gauge), |ξ|²A = A(x)ξ, ξ for ξ ∈ Rⁿ and A ( x ) is an n × n symmetric, uniformly positive definite matrix defined on a bounded domain Ω ⊂ Rⁿ . We also give its L p analogue. As a consequence, we present examples for a standard Laplacian on Rⁿ , Baouendi–Grushin operator, and sub-Laplacians on the Heisenberg group, the Engel group and the Cartan group. Those kind of characterisations for a pair of functions $(W(x),H(x))$ are obtained also for the Rellich inequality. These results answer the open problems of Ghoussoub-Moradifam [16].