A unified divergent approach to Hardy–Poincaré inequalities in classical and variable Sobolev spaces

We present a unified strategy to derive Hardy–Poincaré inequalities on bounded and unbounded domains. The approach allows proving general inequality from which the classical Poincaré Hardy immediately follow. extend idea more context of variable exponent Sobolev spaces. Surprisingly, despite well-known counterexamples Fan et al. (2005) [28] , we show that modular form is actually possible provided one restricts class functions in u ∈ C c ∞ ( Ω ) such | ⩽ 1 . argument, concise constructive, does not require priori knowledge compactness results retrieves geometric information best constants.