A remark on Poincaré inequalities on metric measure spaces - Keith, Stephen; Rajala, Kai (MATHEMATICA SCANDINAVICA, 2004)
We show that, in a complete metric measure space equipped with a doubling Borel regular measure, the Poincaré inequality with upper gradients introduced by Heinonen and Koskela [3] is equivalent to the Poincaré inequality with "approximate Lipschitz constants" used by Semmes in [9].