\(W^{2,1}\) estimate for singular solutions to the Monge-Ampère equation (Q2792162)

The author presents an improved version of his earlier results in [Singular solutions to the Monge-Ampére equation. New York, NY: Columbia University (PhD Thesis) (2015)] and [Commun. Pure Appl. Math. 68, No. 6, 1066--1084 (2015; Zbl 1317.35079)] on interior regularity of a singular (i.e., convex, possibly not strictly convex) solution \(u\) (in the sense of Alexandrov) of the Monge-Ampère equation \(0\leq \lambda \leq \det \left( D^{2}u\right) \leq \Lambda \) in a sub-ball of the unit ball \(B_{1}\) in \(\mathbb{R}^{n}\) (\(n\geq 3\)) subject to the condition \(\left\| u\right\| _{L^{\infty }\left( B_{1}\right) }\leq K\). The main theorem (see also [PhD Thesis, 2015, loc. cit.], Theorem 1.4.3) states that \(\Delta u\in L\log ^{\varepsilon }L\) for small \(\varepsilon >0\) depending on \(n\) in \( B_{1/2}\) and the following a priori estimate holds: \(\int_{B_{1/2}}\Delta u\left( \log \left( 1+\Delta u\right) \right) ^{\varepsilon }dx\leq C\left( n,\lambda ,\Lambda ,K\right) \). In addition, the author constructs an example showing that in general \(u\) is not in \(L\log ^{M}L\) for large \(M\).