If $f(\lambda[\cdot])$ satisfies the structure condition (1.4), then the following holds:

(a) Assume that $p\in(0,n)$. Then there exist constants $c_{p}$, $C_{p}>0$ depending only on $\omega_{X}$, $n$, $p$, $\gamma$ and the generalized entropy ${\mathrm{Ent}}_{p}(F)$ such that

(b) Assume that $p=n$. Then for any $N>0$, there exists constants $c_{N}>0$, $C_{N}>0$ depending on $n,\omega_{X}$, $N$, $\gamma$, and the generalized entropy ${\mathrm{Ent}}_{n}(F)$ so that

(c) We have the energy estimate:

for $N={n\over n-p}$ if $p\in[1,n)$, and for any $N>0$ if $p=n$, where the constant $C$ on the right hand side of (3.6) depends on $n,\omega_{X},\gamma,N$ and the entropy ${\mathrm{Ent}}_{p}(F)$.