Global $$W^{2,p}$$ Regularity on the Linearized Monge-Amp$$\grave{e}$$re Equation with $$\mathrm {VMO}$$ Type Coefficients

In this paper, we establish global $$W^{2,p}$$ estimates for solutions to the linearized Monge-Amp $$\grave{e}$$ re equation $$\begin{aligned} {\mathcal {L}}_{\phi }u:=\mathrm {tr}[\Phi D^2 u]=f \end{aligned}$$ under appropriate conditions on domain, measures, boundary data and f, $$\Phi :=(\mathrm {det}\,D^2\phi )(D^2\phi )^{-1}$$ is cofactor matrix of $$D^2\phi $$ . density measure $$g:=\mathrm satisfies a $$\mathrm {VMO}$$ -type condition.