Functional inequalities on arbitrary Riemannian manifolds

AbstractFor any connected (not necessarily complete) Riemannian manifold, we construct a probability measure of type eV(x)dx, where dx 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