Weighted Poincare inequalities on symmetric convex domains

Abstract

Let α ≥ 0, β ∈ R, 1 ≤ p ≤ q < ∞ with 1 - n / p + n / q , 1 - n + β / p + n + α / q ≥ 0. Let Ω be a bounded convex domain in ℝ n that is symmetric with respect to its center. Define p(x) = dist(x, Ω c ) = inf{|x - γ|: γ ∈ Ω c } and ρ α (E) = ∫ E ρ(x) α dx. Let f be a Lipschitz continuous function on Ω and f Ω,ρ α = ∫ Ω f(x)ρ(x) α dx/ρ α (Ω). We obtain the following weighted Poincare inequality: ∥f - f Ω,ρ α ∥L q ρα (Ω) ≤ Cη β/p-α/q |Ω| 1/q-1/p diam(Ω) 1-β/p+α/q ∥∇f∥ L p ρ β (Ω) where η is the eccentricity of Ω and C is a constant depending only on p, q, α, β, and the dimension n. Moreover, the exponent of η is sharp.