A variant of the Hardy inequality (Q884989)

The inequality of Hardy is proven for Lie groups and some previous results on the same subject are generalized or presented in a simpler form. The author considers a unimodular Lie group \(G\) endowed with a Haar measure and denotes by \(L_{pq}(G) \) the associated Lorentz space. For a given system of vector fields \(X_{1},X_{2},\dots,X_{m}\) the associated sub-Laplacian \(\Delta =-(X_{1}^{2}+\cdots+X_{m}^{2})\) is constructed and the corresponding semigroup \(P_{t}\) is introduced. A proof of the Hardy inequality is given and some comments are added. It is emphasized that the result obtained in this paper can be used to prove a theorem on the existence of solutions to the \(p\)-sub-Laplacian in the case of nilpotent groups with dilatations.