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

Global $W^{2,p}$ regularity on the linearized Monge-Amp$\grave{e}$re equation with $\mathrm{VMO}$ type coefficients