Coercive estimates and integral representation formulas on Carnot groups

For general Carnot groups, we obtain coercive estimates for homogeneous differential operators with constant coefficients, kernels of which have finite dimension. We develop new Sobolev-type integral representations of differentiable functions which are a crucial tool for deriving coercive estimates. Moreover we prove some auxiliary results having independent interest, in particular, Sobolev type embedding and compactness theorems for John domains.