In this paper, we derive a new set of Poincaré inequalities on the sphere, with respect to some Markov kernels parameterized by a point in the ball. When this point goes to the boundary, those Poincaré inequalities are shown to give the curvature-dimension inequality of the sphere, and when it is at the center they reduce to the usual Poincaré inequality. We then extend them to Riemannian manifolds, giving a sequence of inequalities which are equivalent to the curvature-dimension inequality, and interpolate between this inequality and the Poincaré inequality for the invariant measure. This inequality is optimal in the case of the spheres. © 2002 Elsevier Science (USA). All rights reserved.
Scheffer, G. (2003). Local poincaré inequalities in non-negative curvature and finite dimension. Journal of Functional Analysis, 198(1), 197–228. https://doi.org/10.1016/S0022-1236(02)00028-9