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In this paper, we establish a local regularity result for    $$W^{2,p}_{{\mathrm {loc}}}$$  W loc 2 , p solutions to complex degenerate nonlinear elliptic equations    $$F(D^2_{\mathbb {C}}u)=f$$  F ( D C 2 u ) = f when they dominate the Monge–Ampère equation. Notably, we apply our result to the so-called  k -Monge–Ampère equation.

Local regularity for concave homogeneous complex degenerate elliptic equations dominating the Monge–Ampère equation