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In Moser’s famous paper harnack inequality for parabolic equations, he used the following simple Poincare inequality(Lemma 3) $ \int (f(x)-k)^2 w(x) dx \leq c(w) \int |\nabla f|^2 w(x) dx$ where $ k=\int f(x)w(x)dx / \int w(x)dx$ . The weight function $ w(x)$ is just a smooth axially decreasing cut-off function of compact support. Is thisRead more