W2,1 regularity for solutions of the Monge–Ampère equation

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Partial Regularity for Singular Solutions to the Monge‐Ampère Equation (2013, background): “rity or W2;p regularity of singular solutions to detD2u for large p, it is natural to ask what we can show about the integrability of the second derivatives. De Philippis, Figalli and Savin ([8],[7]) recently showed W2;1+regularity of strictly convex solutions to detD2u, where depends only on ; an”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “The answer is yes, as it is shown in [147, 148, 150], and one can even prove sharper estimates (D(2)u is not only L(1) but slightly better, including some L1C" results, also proven by Schmidt in [281]).”
The Monge–Ampère equation and its link to optimal transportation (2013, methodology): “articular Stein’s Theorem implies that if M(D2u)(x) := sup t Z S(x;p;t) jD2uj2L1; then jD2uj2LlogL, that is R jD2ujlog(2 + jD2uj) C. Hence one wants to prove that M(D2u) 2L1, and the key estimate in [36] consists in showing that kM(D 2u)k L 1CkDuk L ; for some constant C= C(n;), which proves the res”