Logarithmic Hardy–Rellich inequalities on Lie groups

In this paper, we establish a family of weighted logarithmic Hardy–Rellich inequalities on connected Lie groups equipped with Hörmander systems of left-invariant vector fields. In the setting of graded Lie groups, we obtain refinements in terms of homogeneous Sobolev norms associated with Rockland operators. As a consequence, on stratified (Carnot) groups we derive Gross-type logarithmic Hardy inequalities with respect to a product measure that is Gaussian on the first stratum and Lebesgue on the higher strata. We also present a logarithmic Poincaré inequality on stratified groups and a fractional logarithmic Hardy inequality for the fractional p-sub-Laplacian on homogeneous groups. Several of the resulting weighted logarithmic Hardy–Rellich inequalities appear to be new already in \(\mathbb {R}^n\) ${\mathbb{R}}^n$ .