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 Lp Rellich type inequality∫Ga|ΔGu|pdx≥∫Gb|u|pdxholds for every u∈C0∞(G) and p> 1. It is worthwhile 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 Lp Rellich type inequality that connects first to second order derivatives and some new two-weight Rellich type inequalities with remainders on bounded domains Ω in G via a differential inequality and the improved two-weight Hardy inequality in Goldstein et al. (Discret Contin Dyn Syst 37:2009–2021, 2017).

Publication Title

Monatshefte fur Mathematik