Continuity of solutions to complex Hessian equations on compact Hermitian manifolds

Abstract

Let $(X,\omega )$ be a compact Hermitian manifold of dimension $n$. We derive an $L^{\infty }$-estimate for bounded solutions to the complex $m$-th Hessian equations on $X$, assuming a positive right-hand side in the Orlicz space $L^{\frac{n}{m}}(\log L{)}^n(h\circ \log \circ \log L{)}^n$, where the associated weight satisfies Kołodziej's Condition. Building upon this estimate, we then establish the existence of continuous solutions to the complex Hessian equation under the prescribed assumptions.