On Hardy and Rellich type inequalities for an Engel-type operator

Abstract

In the paper by Ruzhansky and Suragan it was demonstrated that improved versions of Hardy and Rellich inequalities as well as of uncertainty principles for sums of squares of vector fields on bounded sets of smooth manifolds under certain assumptions on the vector fields, in particular, the obtained results were valid for sums of squares of vector fields on Euclidean spaces and for sub-Laplacians on stratified Lie groups. In this paper we present versions of Hardy and Rellich type inequalities for an Engel-type operator by using properties of the fundamental solution. The obtained results remain unchanged for higher step Engel-type operators.