Interior Hölder estimate for the linearized complex Monge–Ampère equation

Let \(w_0\) be a bounded, \(C^3\), strictly plurisubharmonic function defined on \(B_1\subset \mathbb {C}^n\). Then \(w_0\) has a neighborhood in \(L^{\infty }(B_1)\). Suppose that we have a function \(\phi \) in this neighborhood with \(1-\varepsilon \le MA(\phi )\le 1+\varepsilon \) and there exists a function u solving the linearized complex Monge–Amp\(\grave{\text {e}}\)re equation: \(det(\phi _{k\bar{l}})\phi ^{i\bar{j}}u_{i\bar{j}}=0\). Then there exist constants \(\alpha >0\) and C such that \(|u|_{C^{\alpha }(B_{\frac{1}{2}}(0))}\le C\), where \(\alpha >0\) depends on n and C depends on n and \(|u|_{L^{\infty }(B_1(0))}\), as long as \(\epsilon \) is small depending on n. This partially generalizes Caffarelli–Gutierrez’s estimate for linearized real Monge–Amp\(\grave{\text {e}}\)re equation to the complex version.