  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.
