Interior Hölder estimate for the linearized complex Monge–Ampère equation

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