$W^{2,1}$ regularity for solutions of the Monge-Amp\`ere equation
  In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Amp\`ere 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 $k\in \mathbb N$. 
Guido De Philippis, Alessio Figalli
math.AP