Gz. Lu et Rl. Wheeden, High order representation formulas and embedding theorems on stratified groups and generalizations, STUD MATH, 142(2), 2000, pp. 101-133
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.