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Interior second derivative estimates for solutions to the linearized Monge--Ampère equation

Let $Ω\subset \R^n$ be a bounded convex domain and $ϕ\in C(\barΩ)$ be a convex function such that $ϕ$ is sufficiently smooth on $\partialΩ$ and the Monge--Ampère measure $\det D^2ϕ$ is bounded away from zero and infinity in $Ω$. The corresponding linearized Monge--Ampère equation is \[ \trace(ΦD^2 u) =f, \] where $Φ:= \det D^2 ϕ~ (D^2ϕ)^{-1}$ is the matrix of cofactors of $D^2ϕ$. We prove a conjecture in \cite{GT} about the relationship between $L^p$ estimates for $D^2 u$ and the closeness between $\det D^2ϕ$ and one. As a consequence, we obtain interior $W^{2,p}$ estimates for solutions to such equation whenever the measure $\det D^2ϕ$ is given by a continuous density and the function $f$ belongs to $L^q(Ω)$ for some $q> \max{\{p,n\}}$.