$C^{1,\frac{1}{3}-}$ very weak solutions to the two dimensional Monge-Ampére equation

For any $\theta <\frac{1}{3}$, we show that very weak solutions to the two-dimensional Monge-Ampère equation with regularity $C^{1,\theta }$ are dense in the space of continuous functions. This result is shown by a convex integration scheme involving a subtle decomposition of the defect at each stage. The decomposition diagonalizes the defect and, in addition, incorporates some of the leading-order error terms of the first perturbation, effectively reducing the required amount of perturbations to one.