Fix $1\leq k<n$, and consider the family (5.4) of Hessian equations with respect to the degenerating background metrics. Assume that $\chi$ is big. Then for any $q>{n\over k}$, $\|\varphi\|_{L^{\infty}}$ is bounded uniformly by a constant $C$ depending only on $n,k,q$, $\omega_{X},\chi$ and an upper bound for $\|e^{nF_{t}}\|_{L^{q}(\omega_{X}^{n})}$.