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.
Categories: math.AP