Interior Gradient Estimates for Solutions to the Linearized Monge–ampère Equation

Let φ be a convex function on a strictly convex domain Ω ⊂ Rn, n ≥ 1. The corresponding linearized Monge–Ampère equation is trace(ΦD2u) = f , where Φ := det D2φ (D2φ)−1 is the matrix of cofactors of D2φ. We establish interior Hölder estimates for derivatives of solutions to the equation when the function f on the right hand side belongs to Lp(Ω) for some p > n. The function φ is assumed to be such that φ ∈ C(Ω̄) with φ = 0 on ∂Ω and the Monge–Ampère measure det D2φ is given by a density g ∈ C(Ω) which is bounded away from zero and infinity.