Hardy and Rellich Inequalities with Bessel Pairs

Abstract In this paper, we establish suitable characterisations for a pair of functions (W (x), H (x) ) on a bounded, connected domain Rⁿ in order to have the following Hardy inequality: equation* _ W (x) | u|A² dx _ | d|²AH (x) |u|² dx, \, \, \, u C^1₀ (), equation* where d (x) is a suitable quasi-norm (gauge), ||²A = A (x), for Rⁿ and A (x) is an n × n symmetric, uniformly positive definite matrix defined on a bounded domain 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.