A new sharp Sobolev inequality between the Q-curvature and the scalar curvature on Euclidean space

We establish a sharp Sobolev-type inequality on Rⁿ, n≥ 5, that interpolates between the second-order energy and a weighted first-order term. Specifically, we determine the sharp constant S4,2 in the inequality {equation*} ∫R^n (Δu)^2\,dx ≥ S4,2 (∫R^n|∇ u|²|u|4/n-4\,dx)n-4/n-2, {equation*} for any u∈ H²(Rⁿ), and we classify the cases of equality. This solves the problem of minimizing the total Q-curvature of among metrics on the sphere, conformal to the standard metric, with fixed total scalar curvature.