Improved Poincaré inequalities with weights ✩

In this paper we prove that if Ω ∈ Rn is a bounded John domain, the following weighted Poincaré-type inequality holds: inf a∈R f (x)−a Lq(Ω,w1) C ∇ f (x)d(x)α Lp(Ω,w2) where f is a locally Lipschitz function on Ω, d(x) denotes the distance of x to the boundary of Ω, the weights w1, w2 satisfy certain cube conditions, and α ∈ [0, 1] depends on p, q and n. This result generalizes previously known weighted inequalities, which can also be obtained with our approach