Improved Poincaré inequalities with weights

In this paper we prove that if Ω∈RnΩ∈Rn is a bounded John domain, the following weighted Poincaré-type inequality holds:infa∈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)d(x) denotes the distance of x to the boundary of Ω  , the weights w1,w2w1,w2 satisfy certain cube conditions, and α∈[0,1]α∈[0,1] depends on p,qp,q and n. This result generalizes previously known weighted inequalities, which can also be obtained with our approach.