A note on the PDE approach to the L ∞ $L^\infty$ estimates for complex Hessian equations on transverse Kähler manifolds

In this note, the partial differential equation (PDE) approach of Guo–Phong–Tong and Guo–Phong–Tong–Wang adapted to prove an $L^{\infty }$ estimate for transverse complex Monge–Ampère equations on homologically orientable transverse Kähler manifolds. As an application, a purely PDE-based proof of the regularity of Calabi–Yau cone metrics on $Q$-Gorenstein $T$-varieties is obtained.