In this section we obtain $W^{2,1}$ regularity for singular solutions to the Monge-Ampère equation. Furthermore, by examining the examples in the previous section we show that we cannot improve this result to $W^{2,1+\epsilon}$ regularity for an $\epsilon$ depending on $\lambda,\Lambda$ and $n$.

The following result of Savin, De Philippis and Figalli (see [DFS]) gives $W^{2,1+\epsilon}$ regularity of solutions to $\lambda\leq\det D^{2}u\leq\Lambda$ in compactly contained sections:

$W^{2,1}$ regularity then follows from our main theorem.

We now examine the integrability of $\Delta u$ for the examples constructed in the previous section. On any ball $B_{r}$, by Hölder’s inequality we have

$$\int_{B_{r}}(\Delta u)^{1+\epsilon}\,dx\geq c(n)r^{-\epsilon n}\left(\int_{B_{r}}\Delta u\,dx\right)^{1+\epsilon}.$$

Recall that the subsolutions $w$ grow like $x_{2}^{\beta}=x_{2}^{1+\frac{\delta}{4-3\delta}}$ in the $x_{2}$ direction, and that these functions touch $u$ by below at any $x\in\Sigma=S\times\{0\}\times(-1,1)^{n-2}$. It follows that

$$\sup_{\partial B_{r}(x)}(u-u(x))\geq r^{\beta}$$

for any $x\in\Sigma$. Applying convexity,

	$\displaystyle\int_{B_{r}(x)}(\Delta u)^{1+\epsilon}\,dx$	$\displaystyle\geq c(n)r^{-\epsilon n}\left(\int_{\partial B_{r}}u_{\nu}\,ds\right)^{1+\epsilon}$
		$\displaystyle\geq c(n)r^{(n+\beta-2)(1+\epsilon)-\epsilon n}$
		$\displaystyle\geq c(n)r^{n-1-\epsilon+(1+\epsilon)\frac{\delta}{3}}.$

Fix $\eta$ small and cover $S\times\{0\}\times(-1,1)^{n-2}$ with balls of radius $r_{i}<\eta$. Take a Vitali subcover $\{B_{r_{i}}\}_{i=1}^{\infty}$. It follows that

$$\int_{B_{1}}(\Delta u)^{1+\epsilon}\,dx\geq c(n)\sum_{i=1}^{\infty}r_{i}^{n-1-\epsilon+(1+\epsilon)\frac{\delta}{3}}.$$