On global $W^{2,\delta}$ estimates for the Monge-Amp\`ere equation on general bounded convex domains

Abstract

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