Hardy and Rellich inequalities for anisotropic p-sub-Laplacians

Abstract

In this paper we establish the subelliptic Picone type identities. As consequences, we obtain Hardy and Rellich type inequalities for anisotropic p-sub- Laplacians which are operators of the form $${\mathcal{L}}_{p}f:=\sum _{i=1}^N{X}_i(|X_{i}f{|}^{p_i-2}{X}_{i}f),1<p_i<\infty ,$$ where $X_i,i=1,\dots ,N,$ are the generators of the first stratum of a stratified (Lie) group. Moreover, analogues of Hardy type inequalities with multiple singularities and many-particle Hardy type inequalities are obtained on stratified groups.