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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.

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