First we derive the estimate for $\int_{M}e^{qF}dvol_{g}$ with $q>1$.

From Theorem 5.2, we know

(5.7)

$$-\alpha\psi\geq\frac{\alpha}{\varepsilon}\big(F-2(1+\max_{M}|Ric|)\varphi-C_{5.1}\big).$$

hence

(5.8)

$$\begin{split}C_{5}\geq\int_{M}e^{-\alpha\psi}dvol_{g}\geq&\int_{M}\exp\big(\frac{\alpha}{\varepsilon}(F-2(1+\max_{M}|Ric|)\varphi-C_{5.1})\big)dvol_{g}\\ &\geq\int_{M}\exp\big(\frac{\alpha}{\varepsilon}(F-C_{5.1})\big)dvol_{g}.\end{split}$$

The last inequality holds because we normalized $\varphi$ so that $\varphi\leq 0$. Choose $\varepsilon=\frac{\alpha}{q}$, then we immediately get the desired estimate for $\int_{M}e^{qF}dvol_{g}$. The claimed estimate for $\varphi$ and $\psi$ immediately follows from the estimate for $||e^{F}||_{L^{q}}(q>2)$, given in the lemma below. ∎