Local regularity of the complex Monge-Ampère equation (Q2786829)

The following local regularity result for the complex Monge-Ampère equation NEWLINE\[NEWLINE (dd^c u)^n =f dV NEWLINE\]NEWLINE in \(\mathbb C ^n\) is shown. Let \(u\) be a continuous weak solution in the unit ball \(B\) and let \(f\) be of class \(C^{\alpha}\) and strictly positive (bounded away from zero) there. Then assuming that NEWLINE\[NEWLINE \Delta u \in \mathrm{BMO}(B), NEWLINE\]NEWLINEthe function \(u\) has bounded \(C^{2, \alpha}\) norm in a smaller ball, with a bound depending on NEWLINE\(n\), \(\alpha\), \(\inf f\), \(||\Delta u||_{\mathrm{BMO}}\), \(||f||_{C^\alpha }\).NEWLINENEWLINEFor \(f\in C^{1, 1}\) this statement was obtained by \textit{Z. Błocki} and \textit{S. Dinew} [Math. Ann. 351, No. 2, 411--416 (2011; Zbl 1237.32006)].NEWLINENEWLINEFor the entire collection see [Zbl 1322.53004].