Integral representations and the generalized Poincaré inequality on Carnot groups

We give some integral representations of the form $f(x)=P(f)+K(\nabla f)$ on two-step Carnot groups, where $P(f)$ is a polynomial and $K$ is an integral operator with a specific singularity. We then obtain the weak Poincaré inequality and coercive estimates as well as the generalized Poincaré inequality on the general Carnot groups.