POINCARE INEQUALITIES, ISOPERIMETRIC ESTIMATES, AND REPRESENTATION FORMULAS ON PRODUCT-SPACES

We derive Sobolev-Poincare inequalities for product balls in case the component spaces are metric spaces associated with appropriate collect ions of vector fields on Euclidean space. We also consider weighted ve rsions of such inequalities, representation formulas which express a f unction in terms of integrals of potential type of its component vecto r field gradients in product spaces, and isoperimetric estimates invol ving product balls.