Global $W^{1,p}$ estimates for solutions to the linearized   Monge--Amp\`ere equations
  In this paper, we investigate regularity for solutions to the linearized Monge-Amp\`ere equations when the nonhomogeneous term has low integrability. We establish global   $W^{1,p}$ estimates for all $p<\frac{nq}{n-q}$ for solutions to the equations with right hand side in $L^q$ where $n/2<q\leq n$. These estimates hold under natural assumptions on the domain, Monge-Amp\`ere measures and boundary data. Our estimates are affine invariant analogues of the global $W^{1,p}$ estimates of N. Winter for fully nonlinear, uniformly elliptic equations. 
Nam Q. Le and Truyen Nguyen
math.AP