  The PDE approach developed earlier by the first three authors for $L^\infty$
estimates for fully non-linear equations on K\"ahler manifolds is shown to
apply as well to Monge-Amp\`ere and Hessian equations on nef classes. In
particular, one obtains a new proof of the estimates of
Boucksom-Eyssidieux-Guedj-Zeriahi and Fu-Guo-Song for the Monge-Amp\`ere
equation, together with their generalization to Hessian equations.
