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 $$\begin{aligned} {\mathcal {L}}_{p}f:= \sum _{i=1}^{N} X_i\left( |X_i f|^{p_i-2} X_i f \right) ,\quad 1<p_i<\infty , \end{aligned}$$where \(X_i\), \(i=1,\ldots , N\), 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.