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

Abstract In this note, the partial differential equation (PDE) approach of Guo–Phong–Tong and Guo–Phong–Tong–Wang adapted to prove an 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 ‐Gorenstein ‐varieties is obtained.