In this section, we consider applications of Theorem 2 to fully non-linear equations besides Monge-Ampère equations. For these applications, we need to consider the relative volumes $c_{t}^{n}V_{t}^{-1}$ and the energies $E_{t}(\varphi_{t})$. We begin with a simple estimate for $E_{t}(\varphi_{t})$, which generalizes the simple considerations which applied earlier to Monge-Ampère equations and is a straightforward application of the Hölder inequality,

Next we note the following uniform $L^{p}$ estimate for general functions $u$, whose Hessian is in a cone $\Gamma_{k}=\{\lambda;\sigma_{\ell}>0,\ 1\leq\ell\leq k\}$, which is an analogue of the $\alpha$-invariant estimate for plurisubharmonic functions. It is well-known to experts, but we supply a statement and proof without pluripotential theory, as we could not find a convenient reference:

Fix an $s>0$ and a small $\epsilon>0$. Let $K=\{u\leq-s\}\subset X$ be compact sub-level set of $u$. We choose a sequence of smooth positive functions $\eta_{j}$ with $\int_{X}\eta_{j}\omega_{X}^{n}=1$ which converge to $\eta_{\infty}:=aV_{K}^{2\epsilon-1}\chi_{K}+a\cdot\chi_{X\backslash K}$ in $L^{1+\epsilon}(\omega_{X}^{n})$ and also pointwise, where $V_{K}=\int_{K}\omega_{X}^{n}$ is the volume of the set $K$ and $a>0$ is a constant such that

$$\int_{X}\eta_{\infty}\omega_{X}^{n}=\int_{X}(aV_{K}^{2\epsilon-1}\chi_{K}+a\cdot\chi_{X\backslash K})\omega_{X}^{n}=aV_{K}^{2\epsilon}+a{\mathrm{Vol}(X\backslash K)}=1.$$

It is not hard to see that $1/2\leq a\leq\max(2,2^{2\epsilon})=2$. Hence

$$\int_{X}\eta_{\infty}^{1+\epsilon}\omega_{X}^{n}=a^{1+\epsilon}V_{K}^{\epsilon+2\epsilon^{2}}+a^{1+\epsilon}{\mathrm{Vol}(X\backslash K)}\leq 4.$$

Thus we may assume $\|\eta_{j}\|_{L^{1+\epsilon}(\omega_{X})}\leq 5$ for $j$ large enough. We solve the complex Monge-Ampère equations

$$(\omega_{X}+i\partial\bar{\partial}v_{j})^{n}=\eta_{j}\omega_{X}^{n},\quad\sup_{X}v_{j}=0.$$

By Theorem 1 (or [14]), it holds that $\|v_{j}\|_{L^{\infty}}\leq C_{0}$ for a uniform $C_{0}=C_{0}(n,\omega_{X},\epsilon)$.

By integration by parts we have

			$\displaystyle\int_{X}(-u)(\omega_{X}+i\partial\bar{\partial}v_{j})^{k}\wedge\omega_{X}^{n-k}$		(5.2)
		$\displaystyle=$	$\displaystyle\int_{X}(-u)\omega_{X}\wedge\omega_{v_{j}}^{k-1}\wedge\omega_{X}^{n-k}+(-v_{j})(\omega_{u}-\omega_{X})\wedge\omega_{v_{j}}^{k-1}\wedge\omega_{X}^{n-k}$
		$\displaystyle\leq$	$\displaystyle\int_{X}(-u)\omega_{X}\wedge\omega_{v_{j}}^{k-1}\wedge\omega_{X}^{n-k}+C_{0}\int_{X}\omega_{u}\wedge\omega_{v_{j}}^{k-1}\wedge\omega_{X}^{n-k}$
		$\displaystyle=$	$\displaystyle\int_{X}(-u)\omega_{v_{j}}^{k-1}\wedge\omega_{X}^{n-k+1}+C_{0}.$

Applying (5.2) inductively we get

$$\int_{X}(-u)(\omega_{X}+i\partial\bar{\partial}v_{j})^{k}\wedge\omega_{X}^{n-k}\leq\int_{X}(-u)\omega_{X}^{n}+kC_{0}\leq C,$$

(5.3)

for some uniform constant $C=C(n,k,\omega_{X},\epsilon)>0$. On the other hand, by Newton-Maclaurin inequality that $\big(\frac{\omega_{v_{j}}^{k}\wedge\omega_{X}^{n-k}}{\omega_{X}^{n}}\big)^{1/k}\geq c(n,k)\big(\frac{\omega_{v_{j}}^{n}}{\omega_{X}^{n}}\big)^{1/n}$ we derive from (5.3) that

$$\int_{X}(-u)(\eta_{j})^{k/n}\omega_{X}^{n}\leq C(n,k,\omega_{X},\epsilon).$$

Letting $j\to\infty$ and applying Fatou’s lemma we get

$$\int_{K}(-u)V_{K}^{(2\epsilon-1)k/n}\omega_{X}^{n}\leq C(n,k,\omega_{X},\epsilon)$$

from which we obtain that

$$V_{K}={\mathrm{Vol}}(\{u\leq-s\})\leq C(n,k,\omega_{X},\epsilon)s^{-\frac{n}{(2\epsilon-1)k+n}}.$$

For any $p<\frac{n}{n-k}$, we have

$$\int_{X}(-u)^{p}\omega_{X}^{n}\leq 1+pC(n,k,\omega_{X},\epsilon)\int_{1}^{\infty}s^{p-1-\frac{n}{(2\epsilon-1)k+n}}ds\leq C(n,k,\omega_{X},p)$$

if $\epsilon=\epsilon(p)>0$ is chosen small enough so that the integral above is integrable. The proof of Lemma 8 is complete.

Returning to the applications of Theorem 2, we observe that the condition that $p<{n\over n-k}$ is equivalent to the dual exponent $q$ satisfying $q>{n\over k}$. Thus Theorem 2 combined with Lemmas 7 and 8 imply at once:

We illustrate this theorem by specializing now to the case of Hessian equations, where $f(\lambda)=\sigma_{k}(\lambda)^{1\over k}$ for some $1\leq k\leq n$. The more familiar form of this equation is

$$(\omega_{t}+i\partial\bar{\partial}\varphi_{t})^{k}\wedge\omega_{X}^{n-k}=c_{t}^{k}e^{kF_{t}}\omega_{X}^{n},\quad\sup_{X}\varphi_{t}=0,$$

(5.4)

and the condition $\lambda[\omega_{t,\varphi_{t}}]\in\Gamma_{k}$ is part of the equation²² 2 We remark that usually in the equation (5.4), one normalizes the function $F_{t}$ such that $\int_{X}e^{kF_{t}}\omega_{X}^{n}=V$. However, our normalization is that $\int_{X}e^{nF_{t}}\omega_{X}^{n}=V$.. Thus Theorem 5 applies and, assuming uniform bounds for $\|e^{nF}\|_{L^{q}}$ for some $q>n/k$, it reduces the uniform estimates for $\varphi_{t}$ to a uniform estimate for the relative volumes $c_{t}^{n}/V_{t}$. An important geometric case when the relative volumes can be controlled is when the classe $\chi$ is big, in the sense that its volume $[\chi^{n}]=\int_{X}\chi^{n}$ is strictly positive. In this case, we obtain

The factors $c_{t}$ are determined by

$$c_{t}^{k}\int_{X}e^{kF_{t}}\omega_{X}^{n}=\int_{X}\omega_{t}^{k}\wedge\omega_{X}^{n-k}=O(1).$$

(5.5)

To estimate $c_{t}$, we still need a uniform lower bound of $\int_{X}e^{kF_{t}}\omega_{X}^{n}$. We use Hölder’s inequality as before. Thus we write

$$\int_{X}e^{kF_{t}}\omega_{X}^{n}\geq\int_{\{F_{t}\leq 0\}}e^{nF_{t}}\omega_{X}^{n}+\int_{\{F_{t}>0\}}\omega_{X}^{n}.$$

(5.6)

Recall that by our normalization on $F_{t}$

$$V=\int_{X}e^{nF_{t}}\omega_{X}^{n}=\int_{\{F_{t}\leq 0\}}e^{nF_{t}}\omega_{X}^{n}+\int_{\{F_{t}>0\}}e^{nF_{t}}\omega_{X}^{n}.$$

(5.7)

If the first term on the right hand side of (5.7) is greater than $V/2$, then (5.6) shows that $\int_{X}e^{kF_{t}}\omega_{X}^{n}\geq V/2$; otherwise the second term in (5.7) is greater than $V/2$. Then we have by the Hölder inequality

$$\int_{\{F_{t}>0\}}e^{nF_{t}}\omega_{X}^{n}\leq\Big(\int_{X}e^{qnF_{t}}\Big)^{1/q}\Big(\int_{\{F_{t}>0\}}\omega_{X}^{n}\Big)^{\frac{q-1}{q}}$$

which yields a uniform lower bound of the second term in (5.6) depending additionally on the assumed $\|e^{nF_{t}}\|_{L^{q}}$. Therefore we conclude from (5.5) that $c_{t}\leq C$ for a uniform constant $C>0$. The proof of Theorem 6 is complete.