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 Ko{\l}odziej's Condition. Building upon this estimate, we then establish the existence of continuous solutions to the complex Hessian equation under the prescribed assumptions.