A general approach to weighted Rellich type inequalities on Carnot groups

We give a simple sufficient criterion on a pair of nonnegative weight functions a and b on a Carnot group G, so that the following general weighted L-p Rellich type inequality integral(G) a vertical bar Delta(G)u vertical bar(p) dx >= integral(G) b vertical bar u vertical bar(p) dx holds for every u is an element of C-0(infinity)(G) and p > 1. It is worth while to notice that our method easily derives previously known weighted Rellich type inequalities with a sharp constant in a more adequate fashion and also enables us to obtain new ones. We also present a sharp L-p Rellich type inequality that connects first to second order derivatives and some new two-weight Rellich type inequalities with remainders on bounded domains Omega in G via a differential inequality and the improved two-weight Hardy inequality in Goldstein et al.