It is well known that in a given Kähler class, cscK metrics is global minimizer of the K-energy functional, by the main result of [1]. In particular, it follows that the K energy functional of $\varphi$ is a priori bounded from above. Recall the decomposition formula for K energy functional $E$, proved in [7]:

(5.22)

$$K(\varphi)=\int_{M}\log\frac{\omega_{\varphi}^{n}}{\omega_{0}^{n}}\frac{\omega_{\varphi}^{n}}{n!}+J_{-Ric}(\varphi).$$

In the above, $J_{-Ric}$ is defined in terms of its derivative, namely

$$\frac{dJ_{-Ric}}{dt}=\int_{M}\frac{\partial\varphi}{\partial t}(-tr_{\varphi}Ric+\underline{R})\frac{\omega_{\varphi}^{n}}{n!}.$$

It is well known in the literature that $J_{-Ric}$ can be bounded in terms of $C^{0}$ norm of the potential function $\varphi.\;$ A bound for $\int_{M}e^{F}|F|dvol_{g}$ follows from here.

Now we prove the second part of the theorem. First Corollary 5.4 gives a bound for $F$ from above and Corollary 5.2 gives a bound for $||\varphi||_{0}$. Proposition 2.1 gives a bound for $F$ from below. ∎