On global \(W^{2,\delta }\) estimates for the Monge–Ampère equation on general bounded convex domains

We establish global \(W^{2,\delta }\) $W^{2,\delta }$ estimates, for all \(\delta <\frac{1}{n-1}\) $\delta <\frac{1}{n-1}$ , for convex solutions to the Monge–Ampère equation with positive \(C^{2,\beta }\) $C^{2,\beta }$ right-hand side and zero boundary values on general bounded convex domains in \(\mathbb R^n\) ${\mathbb{R}}^n$ \((n\ge 2).\) $(n\ge 2).$ We exhibit examples showing that global \(W^{2, \frac{n}{2(n-1)}}\) $W^{2,\frac{n}{2(n-1)}}$ estimates fail in all dimensions, so the range of \(\delta \) $\delta$ is sharp in two dimensions.