High order representation formulas and embedding theorems on stratified groups and generalizations

We derive various integral representation formulas for a function minus a p olynomial in terms of vector field gradients of the function of appropriate ly high order. Our results hold in the general setting of metric spaces, in cluding those associated with Carnot-Caratheodory vector fields, under the assumption that a suitable L-1 to L-1 Poincare inequality holds. Of particu lar interest are the representation formulas in Euclidean space and stratif ied groups, where polynomials exist and L-1 to L-1 Poincare inequalities in volving high order derivatives are known to hold. We apply the formulas to derive embedding theorems and potential type inequalities involving high or der derivatives.