Journal of Inequalities and Applications 
Volume 3 (1999), Issue 1, Pages 65-89
doi:10.1155/S1025583499000053

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

Abstract

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.