A general approach to weighted Rellich type inequalities on Carnot groups

Abstract

We give a simple sufficient criterion on a pair of nonnegative weight functions a and b on a Carnot group $$\mathbb {G},$$ so that the following general weighted $$L^{p}$$ Rellich type inequality $$\begin{aligned} \int _{\mathbb {G}}a\left| \Delta _{\mathbb {G}}u\right| ^{p}dx\ge \int _{\mathbb {G}}b\left| u\right| ^{p}dx \end{aligned}$$ holds for every $$u\in C_{0}^{\infty }(\mathbb {G})$$ 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 $$L^{p}$$ 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 $$\mathbb {G}$$ via a differential inequality and the improved two-weight Hardy inequality in Goldstein et al. (Discret Contin Dyn Syst 37:2009–2021, 2017).