Interior $C^{1,\alpha }$ regularity on the linearized Monge-Amp$\grave{e}$re equation with $\mathrm{VMO}$ type coefficients

Abstract

In this paper, we establish interior $C^{1,\alpha }$ estimates for solutions of the linearized Monge-Amp$\grave{e}$re equation $${\mathcal{L}}_{\varphi }u:=\mathrm{tr}[\Phi D^{2}u]=f,$$ where the density of the Monge-Amp$\grave{e}$re measure $g:=\det D^2\varphi$ satisfies a $\mathrm{VMO}$-type condition and $\Phi :=(\det D^2\varphi )(D^2\varphi {)}^{-1}$ is the cofactor matrix of $D^2\varphi$.

9 pages