We prove and give numerical results for two lower bounds and eleven upper bounds to the optimal constant k0 = k0(n,α) in the inequality ∥u∥2n/(n-2α) ≤ k0 ∥∇u∥α2 ∥u∥1-α 2, u ∈ H
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We prove and give numerical results for two lower bounds and eleven upper bounds to the optimal constant k0 = k0(n,α) in the inequality ∥u∥2n/(n-2α) ≤ k0 ∥∇u∥α2 ∥u∥1-α 2, u ∈ H1(ℝn), for n = 1, 0 < α ≤ 1/2, and n ≥ 2, 0 < α < 1. This constant k0 is the reciprocal of the infimum λn,α for u ∈ H1(ℝn) of the functional Λn,α = ∥∇u∥α2 ∥u∥1-α 2/∥u∥2n/(n-2α), u ∈ H1(ℝn), where for n = 1, 0 < α ≤ 1/2, and for n ≥ 2, 0 < α < 1. The lowest point in the point spectrum of the Schrödinger operator τ = -Δ+q on ℝn with the real-valued potential q can be expressed in λn,α for all q _ = max(0,-q) ∈ Lp(ℝn), for n = 1, 1 ≤ p < ∞, and n ≥ 2, n/2 < p < ∞, and the norm ∥q _ ∥p.
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