Journal of Inequalities and Applications
Volume 3 (1999), Issue 1, Pages 65-89

Some remarks about Poincaré type inequalities and representation formulas in metric spaces of homogeneous type

Bruno Franchi1 and Richard L. Wheeden2

1Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 5, Bologna 140127, Italy
2Department of Mathematics, Rutgers University, New Brunswick 08903, NJ, USA

Received 9 October 1997; Revised 29 December 1997

Copyright © 1999 Bruno Franchi and Richard L. Wheeden. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We derive an integral representation formula for a function in terms of its vector field gradient, assuming a less restrictive growth condition on the volumes of balls than was previously known. We give the explicit form of the constants involved in the formula. We also show that the required growth condition is satisfied by a large class of Carnot– Carathéodory vector fields.