Sharp conditions for weighted 1-dimensional Poincaré inequalities

Abstract

We obtain necessary and sufficient conditions for all weighted one dimensional Poincaré inequalities. Weighted Hardy-type inequalities are an example of our estimates. We use these estimates to obtain some Poincaré type inequalities on product spaces for product weights. In some cases, we also obtain the exact constants in the inequalities. Finally, we apply our result to study solutions of ordinary differential equations with given average value.