A Priori Estimates of the Prescribed Scalar Curvature on the Sphere

Resumen

This paper considers the prescribed scalar curvature problem on the sphere for n ≥ 3. Given a prescribed scalar curvature function K : Sn → R and a centered dilation defined by Fy = Σ−1 ◦ Dβ ◦ Σ, y ∈ Bn+1, where Σ is the stereographic projection and Dβ is a dilation in Rn, in this work we estimate the gradient of the function K near the critical point of the function Jp(y) = Sn K(ζ)φp+1dσ(ζ) where φ(y) = |(F −1 y ) | n−2 2 . We will use this estimate to find Lp estimates of the first two y-derivatives of the function K ◦ Fy(ξ). K