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

Abstract

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

9 pages