$W^{2,1}$ regularity for solutions of the Monge-Ampère equation

In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Ampère equation, with right hand side bounded away from zero and infinity, is $W_{\rm loc}^{2,1}$. This is obtained by showing higher integrability a-priori estimates for $D^2 u$, namely $D^2 u \in L\log^k L$ for any $k\in \mathbb N$.

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