Uniform $L^{\infty }$ estimates for complex hessian equations on compact Hermitian manifolds

Abstract

We develop a pluripotential approach to complex Hessian equations on compact Hermitian manifolds. In this setting, the lack of closedness of the background metric introduces torsion terms that prevent a direct extension of the Kähler theory. Our main result is a uniform $L^{\infty }$ estimate for bounded $\omega$-$m$-subharmonic solutions of the equation \[ (ω+ dd^c u)^m \wedge ω^{n-m} = cf\,ω^n, \] under the assumption that $f\in L^p$, $f\ge 0$ for some $p>1$. The proof combines a weak comparison principle with torsion error, a capacity theory adapted to the Hermitian setting, and a nonlinear iteration scheme controlling the decay of sublevel sets. As applications, we obtain existence, stability and compactness results for weak solutions with $L^p$ densities. These results extend several aspects of the pluripotential theory. of complex Hessian equations beyond the Kähler framework.