Hardy and Rellich inequalities for anisotropic p-sub-Laplacians

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 $$\mathcal{L}_p f := \sum_{i=1}^{N}X_i(|X_if|^{p_i-2}X_i f), \quad 1<p_i< \infty,$$ 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.