Gz. Lu et Rl. Wheeden, POINCARE INEQUALITIES, ISOPERIMETRIC ESTIMATES, AND REPRESENTATION FORMULAS ON PRODUCT-SPACES, Indiana University mathematics journal, 47(1), 1998, pp. 123-151
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.