Fully non-linear degenerate elliptic equations in complex geometry

Abstract: We derive an a priori real Hessian estimate for solutions of a large family of geometric fully non-linear elliptic equations on compact Hermitian manifolds, which is independent of a lower bound for the right-hand side function. This improves on the estimates of Sz\'ekelyhidi and additionally applies to elliptic equations with a degenerate right-hand side. As an application, we establish the optimal $C^{1,1}$ regularity of envelopes of $(\theta, m)$-subharmonic functions on compact Hermitian manifolds.