Functional inequalities on arbitrary Riemannian manifolds

Abstract

For any connected (not necessarily complete) Riemannian manifold, we construct a probability measure of type e V ( x ) d x , where d x is the Riemannian volume measure and V is a function C ∞ -smooth outside a closed set of zero volume, satisfying Poincaré–Sobolev type functional inequalities. In particular, V is C ∞ -smooth on the whole manifold when the Poincaré and the super-Poincaré inequalities are considered. The Sobolev inequality for infinite measures are also studied.