Research

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.

Let $(X, \omega)$ be a compact Hermitian manifold of dimension $n$. We show that all $(\omega ,m)$-subharmonic functions are $L^p$-integrable on $X$, for any $ p < \frac{n}{n-m}$.