In this section, we illustrate the versatility of our approach by establishing also inequalities of Trudinger type for general non-linear energies. Let $f(\lambda)$ be a fully nonlinear operator satisfying the same structural conditions as in Theorem 1, and define for each $p>0$,

(this notation is slightly different from the notation $E_{t}$ used earlier for degenerating background metrics, but there should be no confusion, as the background metric is here fixed, and the index $t$ can be dropped). Recall that we had established before integral estimates for $\varphi$ in terms of the entropy ${\rm Ent}_{p}$. Trudinger estimates are also exponential estimates, but in terms of the energy $E_{p}$. We have

We note that when specialized to the case when $f(\lambda)$ is the Monge-Ampère operator $f(\lambda)=(\prod_{k=1}^{n}\lambda_{k})^{\frac{1}{n}}$, our theorem recovers an inequality proved in [2, 10]. Moreover, our estimate has the major advantage that all constants there depend only on the $\alpha$-invariant of the underlying manifold, hence is uniform over degenerating families with uniform $\alpha$-invariants.