# Uniform $L^{\infty}$ estimates: subsolutions to fully nonlinear partial
differential equations

Bin Guo and Duong H. Phong  Department of Mathematics & Computer Science,
Rutgers University, Newark, NJ 07102<EMAIL_ADDRESS>Department of
Mathematics, Columbia University, New York, NY 10027<EMAIL_ADDRESS>
###### Abstract.

Uniform bounds are obtained using the auxiliary Monge-Ampère equation method
for solutions of very general classes of fully non-linear partial differential
equations, assuming the existence of a ${\mathcal{C}}$-subsolution in the
sense of G. Székelyhidi and B. Guan. The main advantage over previous
estimates is in the Kähler case, where the new estimates remain uniform as the
background metric is allowed to degenerate.

Work supported in part by the National Science Foundation under grants
DMS-2203273, and DMS-2303508, and the collaboration grant 946730 from Simons
Foundation.

##  1\. Introduction

This paper is motivated by two relatively recent developments. The first is
the new theory of ${\mathcal{C}}$-subsolutions for fully non-linear equations
on compact Hermitian manifolds due to G. Székelyhidi [24] and B. Guan [9], a
parabolic version of which was subsequently developed in [20]. Subsolutions
are of great importance for non-linear equations, as their existence are
clearly a necessary condition for the solvabiliy of the equations themselves.
A significant difficulty for equations on compact manifolds was that the naive
notion may only allow trivial subsolutions, and this difficulty was only
overcome recently with the notion of ${\mathcal{C}}$-subsolution introduced in
[24, 9, 20]. The second development is the new auxiliary Monge-Ampère equation
method for $L^{\infty}$ estimates introduced by the authors in joint work with
F. Tong for equations respectively on Kähler manifolds [14] and Hermitian
manifolds [10]. A great advantage of these new methods is that they lead to
$L^{\infty}$ estimates which are uniform with respect to large classes of
background metrics, the power of which can be appreciated from many recent
applications in complex geometry [4, 13, 11, 12]. A natural question is then
whether the existence of subsolutions can lead, not just $L^{\infty}$
estimates, but $L^{\infty}$ estimates with uniformity comparable to those in
[13, 11]. The goal of the present paper is to provide a positive answer to
this question.

We begin by providing a general formulation of the equations which we shall
consider. Let $X$ be a complex manifold of complex dimension $n$ with a
Hermitian metric $\omega_{X}$. Let $\omega$ be another Hermitian metric and
$\chi$ a closed $(1,1)$-form on $X$. For a smooth function $\varphi$, let
$h_{\varphi}=\omega^{-1}\cdot\chi_{\varphi}:TX\to TX$ be the endomorphism
defined by $\omega$ and
$\chi_{\varphi}=\chi+\sqrt{-1}\partial\overline{\partial}\varphi$, and let
$\lambda[h_{\varphi}]$ is the (unordered) vector of eigenvalues of this
endomorphism. We consider the following fully nonlinear equation on $X$:

(1.1) $\displaystyle
f(\lambda[h_{\varphi}])=e^{F},\quad{\mathrm{sup}}_{X}\varphi=0,$

where $e^{F}$ is a positive smooth function normalized such that

$\int_{X}e^{nF}\omega^{n}=[\omega]^{n}=\int_{X}\omega^{n}.$

The nonlinear operator $f:\Gamma\to{\mathbf{R}}_{>0}$ is required to satisfy
the following conditions:

(i): $\Gamma\subset{\mathbf{R}}^{n}$ is an open convex symmetric cone such
that $\Gamma_{n}\subset\Gamma\subset\Gamma_{1}$;

(ii): $\frac{\partial f}{\partial\lambda_{i}}(\lambda)>0$ for any
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\Gamma$, which implies the
ellipticity of the equation (1.1).

In the papers [14, 2], it was further assumed that $f$ satisfies the
structural condition

(1.2) $\prod_{i=1}^{n}\frac{\partial
f}{\partial\lambda_{i}}(\lambda)\geq\hat{\gamma},\quad\forall\lambda\in\Gamma_{n}$

for some non-negative function $\hat{\gamma}$ which is required to be bounded
from below by a fixed positive constant. This is a natural condition, which
was subsequently shown by Harvey and Lawson [17] to be satisfied by very large
classes of equations, including all Garding-Dirichlet operators [16]. It is
also a natural condition arising in the study of $C^{2}$ estimates [2, 6, 5].
However, as stressed in [11], in many important geometric applications,
$\hat{\gamma}$ may vanish along a complex codimension one subvariety. Another
important example of geometric equation which does not obey the condition of
$\hat{\gamma}$ being strictly positive is the $J$ equation, which admits a
solution if and only if a subsolution exists [22, 26].

We shall be interested in obtaining $L^{\infty}$ bounds for solutions of the
equation (1.1) assuming the existence of a subsolution, even in the absence of
the structural condition (1.2) of the operator $f$. The following notion of
$\mathcal{C}$-subsolution is taken from [24] (see also [20]).

######  Definition 1.1.

Given $\delta>0,R>0$, we say a $(1,1)$-form
$\chi^{\prime}=\chi-\sqrt{-1}\partial\overline{\partial}\theta$ is a
$(\delta,R)$-subsolution of the equation (1.1), if the following holds

(1.3)
$(\lambda[h_{\chi^{\prime}}]-\delta{\mathbf{1}}+\Gamma_{n})\cap\partial\Gamma^{e^{F(z)}}\subset
B(0,R)\subset{\mathbb{R}}^{n},\quad\forall z\in X,$

where ${\mathbf{1}}=(1,\ldots,1)\in{\mathbb{R}}^{n}$ is the fixed vector,
$h_{\chi^{\prime}}=\omega^{-1}\cdot\chi^{\prime}$ is the endomorphism defined
by $\chi^{\prime}$ relative to $\omega$, and

$\Gamma^{e^{F(z)}}=\\{\lambda\in\Gamma~{}|~{}f(\lambda)\geq e^{F(z)}\\}.$

The most important case for us is the Kähler case, to be discussed in Section
2, where estimates are obtained which are uniform with respect to large sets
of forms $\chi$ reaching the boundary of the Kähler cone. The Hermitian case
is discussed in Section 3. Finally, in Section 4, we discuss a notion of
subsolution closer in spirit to numerical criteria in stabilty.

##  2\. The Kähler case

In this section, we assume $(X,\omega_{X})$ is a Kähler manifold. We will
derive uniform estimates for $\varphi$ when a subsolution exists. Here uniform
estimates refer to the ones which do not depend on a fixed reference metric,
in particular, it allows the reference metrics to degenerate. For the purpose
of uniform estimates, we need to consider a subset of the space of Kähler
metrics on $X$. As in the joint work [11] with Song and Sturm, for a given
Kähler metric $\omega$, we will denote its relative volume function as

(2.4)
$e^{F_{\omega}}=\frac{1}{V_{\omega}}\frac{\omega^{n}}{\omega_{X}^{n}},\quad
V_{\omega}=\int_{X}\omega^{n}.$

Given constants $A>0,K>0$, $p>n$ and a continuous nonnegative function
$\gamma\in C^{0}(X)$, we consider the following subset of Kähler metrics

$\displaystyle{\mathcal{W}}(A,K,p,\gamma,\omega_{X})$ $\displaystyle:=$
$\displaystyle\Big{\\{}\omega\mbox{ a K\"{a}hler
metric}~{}|~{}[\omega]\cdot[\omega_{X}]^{n-1}\leq A,e^{F_{\omega}}\geq\gamma,$
$\displaystyle\quad\mbox{and
}\,{\mathcal{N}}_{p}(\omega)=\int_{X}|F_{\omega}|^{p}e^{F_{\omega}}\omega_{X}^{n}\leq
K\Big{\\}}.$

We will always assume that $\gamma$ satisfies
${\mathrm{dim}}_{{\mathcal{H}}}\gamma^{-1}(0)<2n-1$. When the parameters are
fixed, we will simply write ${\mathcal{W}}$ as this set.

Suppose $\chi^{\prime}=\chi-\sqrt{-1}\partial\overline{\partial}\theta$ is a
$(\delta,R)$-subsolution to (1.1) as in Definition 1.1, which satisfies the
assumption:

(2.6) $-n\kappa_{1}\leq{\rm tr}_{\omega}\chi^{\prime}\leq\kappa_{2},$

for constants $\kappa_{1}\geq 0,\,\kappa_{2}\geq 0$.

Our main theorem is as follows:

######  Theorem 2.1.

Let the assumptions be given as above, and $\varphi$ is a $C^{2}$ solution to
the equation (1.1), then there exists a constant $C>0$ depending on the given
parameters $n,A,K,p,\gamma$, and $\delta,R,\kappa_{1},\kappa_{2}$ such that

(2.7)
${\mathrm{sup}}_{X}|(\varphi+\theta)-{\mathrm{sup}}_{X}(\varphi+\theta)|\leq
C.$

To prove this theorem, we need the following preliminary lemmas. First, we
recall an estimate on the Green’s function $G_{\omega}$ associated to
$\omega$.

######  Lemma 2.1 ([11, 13]).

There exists a constant $C_{0}>0$ which depends on $p,A,K,\gamma$ and
$\omega_{X}$ such that

(2.8) $\sup_{x\in X}\Big{(}\sup_{y\in X}(-G_{\omega}(x,y))\cdot
V_{\omega}+\int_{X}|G_{\omega}(x,y)|\omega^{n}(y)\Big{)}\leq C_{0}.$

Next we have the following uniform $L^{1}$ and $L^{q}$ bounds:

######  Lemma 2.2.

Let $\omega\in{\mathcal{W}}$ be a Kähler metric, then we have

(2.9) $\frac{1}{V_{\omega}}\int_{X}(-\psi)\omega^{n}\leq
2nC_{0},\quad\forall\psi\in PSH(X,\omega)\mbox{ with }\sup_{X}\psi=0,$

where $C_{0}$ is the constant in Lemma 2.1.

Proof of Lemma 2.2. By smooth approximation we may assume $\psi$ is $C^{2}$
and $\omega$-PSH. We assume $\sup_{X}\psi=\psi(x_{0})=0$ for some $x_{0}\in
X$. Applying Green’s formula to $\psi$ at $x_{0}$ gives

$\psi(x_{0})=\frac{1}{V_{\omega}}\int_{X}\psi\omega^{n}+\int_{X}[G_{\omega}(x_{0},\cdot)-\inf_{z\in
X}G_{\omega}(x_{0},z)](-\Delta_{\omega}\psi)\omega^{n}.$

Hence by Lemma 2.1, we have

$\frac{1}{V_{\omega}}\int_{X}(-\psi)\omega^{n}\leq
n\int_{X}\\{|G_{\omega}(x_{0},\cdot)|+\frac{C_{0}}{V_{\omega}}\\}\omega^{n}\leq
2nC_{0}.$

Q.E.D.

######  Lemma 2.3.

Let $\omega\in{\mathcal{W}}$ be a Kähler metric. For any $q>n$, there exists a
constant $C_{1}=C_{1}(p,A,K,\gamma,\omega_{X},q)>0$ such that for any
$\omega$-PSH function $\psi$ with $\sup_{X}\psi=0$, we have

(2.10) $\frac{1}{V_{\omega}}\int_{X}({-\psi})^{q}\omega^{n}\leq C_{1}.$

Proof of Lemma 2.3. By Proposition 3.1 in [11], the condition
$[\omega]\cdot[\omega_{X}]^{n-1}\leq A$ implies that there exists a smooth
$(1,1)$-form $\tilde{\omega}\in[\omega]$ such that

(2.11) $-C_{2}\omega_{X}\leq\tilde{\omega}\leq C_{2}\omega_{X}$

for some $C_{2}=C_{2}(A,\omega_{X})>0$. By
$\sqrt{-1}\partial\overline{\partial}$-lemma, we can write
$\omega=\tilde{\omega}+\sqrt{-1}\partial\overline{\partial}u$ for an
$\tilde{\omega}$-PSH function $u$ normalized by $\sup_{X}u=0$. Since by (2.11)
it holds that $\tilde{\omega}\leq C_{2}\omega_{X}$, we see that $u$ is
$(C_{2}\omega_{X})$-PSH, hence the $\alpha$-invariant estimates hold for $u$
(see [15, 25]), that is, there exist
$\alpha^{\prime}=\alpha^{\prime}(C_{2},\omega_{X})>0$ and
$C_{3}=C_{3}(C_{2},\omega_{X})>0$ such that

(2.12) $\int_{X}e^{-\alpha^{\prime}u}\omega_{X}^{n}\leq C_{3}.$

For any $\omega$-PSH function $\psi$ with $\sup_{X}\psi=0$, $\psi+u$ is also
$\tilde{\omega}$-PSH, hence $(C_{2}\omega_{X})$-PSH. Again by
$\alpha$-invariant estimates, we have

(2.13) $\int_{X}e^{\alpha^{\prime}(\sup_{X}(\psi+u)-\psi-u)}\omega_{X}^{n}\leq
C_{3}.$

To finish the proof, it suffices to prove a lower bound of $\sup_{X}(\psi+u)$.
We note that by Young’s inequality

(2.14)
$\frac{1}{V_{\omega}}\int_{X}(-u)\omega^{n}=\int_{X}(-u)e^{F_{\omega}}\omega_{X}^{n}\leq
C\int_{X}e^{F_{\omega}}(1+|F_{\omega}|)\omega_{X}^{n}+C\int_{X}e^{-\alpha^{\prime}u}\omega_{X}^{n}\leq
C_{4},$

where we have used (2.12) and $C_{4}>0$ depends on $p,A,K,$ and $\omega_{X}$.
Combining (2.9) and (2.14) yields

$\frac{1}{V_{\omega}}\int_{X}(-u-\varphi)\omega^{n}\leq 2nC_{0}+C_{4}=C_{5}.$

Hence $\sup_{X}(u+\varphi)\geq-C_{5}$ as claimed. Together with (2.13) this
implies that

(2.15) $\int_{X}e^{\alpha^{\prime}(-\psi)}\omega_{X}^{n}\leq
C_{3}e^{C_{5}}=:C_{6}.$

To prove (2.10), we apply Young’s inequality again to get

$(-\psi)e^{F_{\omega}/q}\leq
e^{F_{\omega}/q}(1+|F_{\omega}|^{p/q})+C_{p,q}e^{-\alpha^{\prime}\psi/q}.$

Taking $q$th power on both sides of the above inequality gives

(2.16) $(-\psi)^{q}e^{F_{\omega}}\leq
Ce^{F_{\omega}}(1+|F_{\omega}|^{p})+C_{p,q}e^{-\alpha^{\prime}\psi}.$

Integrating both sides of (2.16) over $X$ against the measure $\omega_{X}^{n}$
yields the desired inequality (2.10), by making use of (2.15). Q.E.D.

######  Lemma 2.4.

Let the assumptions be as above. For any $v\in C^{2}$ with $\sup_{X}v=0$
satisfying
$\lambda[h_{\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}v}]\in\Gamma\subset\Gamma_{1}$,
we have

(2.17) $\frac{1}{V_{\omega}}\int_{X}(-v)\omega^{n}\leq
2\kappa_{2}C_{0}=:C_{7},$

where $C_{0}$ is the constant in Lemma 2.1.

Proof of Lemma 2.4. The condition
$\lambda[h_{\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}v}]\in\Gamma_{1}$
implies that ${\rm tr}_{\omega}\chi^{\prime}+\Delta_{\omega}v>0$. By (2.6)
this gives $\Delta_{\omega}v\geq-\kappa_{2}$. Then we can apply the Green’s
formula for $v$ at its maximum point $x_{0}$ to get

$0\leq\frac{1}{V_{\omega}}\int_{X}v\omega^{n}+\int_{X}[G_{\omega}(x_{0},\cdot)-\inf_{z}G_{\omega}(x_{0},z)]\kappa_{2}\omega^{n}.$

Q.E.D.

Proof of Theorem 2.1. Let $\varphi$ be a $C^{2}$ function solving the equation
(1.1). Denote $\varphi^{\prime}=\varphi+\theta$ and by adding an appropriate
constant to $\varphi^{\prime}$ we may assume $\sup_{X}\varphi^{\prime}=0$. We
have

$\chi+\sqrt{-1}\partial\overline{\partial}\varphi=\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}.$

Hence, we can view $\varphi^{\prime}$ as a solution to (1.1) if $\chi$ is
replaced by $\chi^{\prime}$. As in [14, 10], for any $s\geq 0$, we consider
the sublevel set $\Omega_{s}$ of $\varphi^{\prime}$ by $\Omega_{s}=\\{z\in
X:\varphi^{\prime}(z)<-s\\}$. We solve the following auxiliary complex Monge-
Ampère equation which admits a smooth solution since $\omega$ is a Kähler
metric [27]

(2.18)
$(\omega+\sqrt{-1}\partial\overline{\partial}\psi_{s,k})^{n}=\frac{\tau_{k}(-\varphi^{\prime}-s)}{A_{s,k}}\omega^{n},\quad\sup_{X}\psi_{s,k}=0,$

where the constant $A_{s,k}$ is defined so that the equation is compatible,
i.e.

(2.19)
$A_{s,k}:=\frac{1}{V_{\omega}}\int_{X}\tau_{k}(-\varphi^{\prime}-s)\omega^{n}\to\frac{1}{V_{\omega}}\int_{\Omega_{s}}(-\varphi^{\prime}-s)\omega^{n}=:A_{s},$

as $k\to\infty$. We consider the comparison function

(2.20)
$\Psi=-\varepsilon(-\psi_{s,k}+\Lambda)^{\frac{n}{n+1}}-\varphi^{\prime}-s,$

where

(2.21) $\varepsilon=\beta_{1}A_{s,k}^{\frac{1}{n+1}},\quad\mbox{where
}\beta_{1}:=\Big{(}\frac{n+1}{n}(R+\delta+\kappa_{1})\Big{)}^{\frac{n}{n+1}},$

and $\Lambda$ is given by

(2.22) $\Lambda=\beta_{2}A_{s,k},\quad\mbox{where
}\beta_{2}=\frac{n}{n+1}(\frac{1}{\delta})^{n+1}(R+\delta+\kappa_{1})^{n}.$

We claim that $\Psi\leq 0$ on $X$. By definition, it suffices to check this
when the maximum point of $\Psi$ is achieved at $x_{0}\in\Omega_{s}$. By the
maximum principle, we have $\sqrt{-1}\partial\overline{\partial}\Psi\leq 0$ at
$x_{0}$, which subsequently implies that at $x_{0}$,

$\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}\geq\frac{n\varepsilon}{n+1}(-\psi_{s,k}+\Lambda)^{-\frac{1}{n+1}}\sqrt{-1}\partial\overline{\partial}\psi_{s,k}+\frac{n\varepsilon}{(n+1)^{2}}(-\psi_{s,k}+\Lambda)^{-\frac{n+2}{n+1}}\partial\psi_{s,k}\wedge\bar{\partial}\psi_{s,k}.$

Therefore, adding $\chi^{\prime}$ on both sides of the above we obtain

(2.23)
$\displaystyle\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}$
$\displaystyle\geq$
$\displaystyle\frac{n\varepsilon}{n+1}(-\psi_{s,k}+\Lambda)^{-\frac{1}{n+1}}(\omega+\sqrt{-1}\partial\overline{\partial}\psi_{s,k})$
$\displaystyle+\chi^{\prime}+[-\frac{n\varepsilon}{n+1}(-\psi_{s,k}+\Lambda)^{-\frac{1}{n+1}}]\omega$
$\displaystyle\geq$
$\displaystyle\frac{n\varepsilon}{n+1}(-\psi_{s,k}+\Lambda)^{-\frac{1}{n+1}}(\omega+\sqrt{-1}\partial\overline{\partial}\psi_{s,k})+\chi^{\prime}-\delta\omega,$

where in the last line we have applied the choice of constant $\Lambda$ in
(2.22) and the assumption (2.6). From the inequality (2.23), we readily get
that at the point $x_{0}$,

(2.24)
$\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}=\chi^{\prime}-\delta\omega+\rho,$

for some positive $(1,1)$-form $\rho$. In terms of eigenvalues, (2.24) is
equivalent to

$\lambda[h_{\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}}]\in\lambda[h_{\chi^{\prime}}]-\delta{\mathbf{1}}+\Gamma_{n}.$

Moreover, from the equation (1.1) we also
have$f(\lambda[h_{\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}}])=e^{F(x_{0})}$.
Thus by the definition of a $(\delta,R)$-subsolution $\chi^{\prime}$, we
conclude that the vector of eigenvalues
$\lambda[h_{\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}}]|_{x_{0}}\in
B(0,R)\subset{\mathbb{R}}^{n}$. If we denote

$\lambda[h_{\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}}]|_{x_{0}}=(\lambda_{1},\ldots,\lambda_{n}),$

then $|\lambda_{i}|\leq R$ for each $i=1,\ldots,n$. Taking trace with respect
to $\omega$ on both sides of (2.23), together with the assumption (2.6), we
obtain that

(2.25)
$nR+n\delta+n\kappa_{1}\geq\frac{n\varepsilon}{n+1}(-\psi_{s,k}+\Lambda)^{-\frac{1}{n+1}}{\rm
tr}_{\omega}(\omega+\sqrt{-1}\partial\overline{\partial}\psi_{s,k})>0.$

Applying the arithmetic-geometric inequality to the RHS of (2.25), and using
the auxiliary equation (2.18) satisfied by
$\omega+\sqrt{-1}\partial\overline{\partial}\psi_{s,k}$, we get

(2.26) $\displaystyle(nR+n\delta+n\kappa_{1})$ $\displaystyle\geq$
$\displaystyle\frac{n^{2}\varepsilon}{n+1}(-\psi_{s,k}+\Lambda)^{-\frac{1}{n+1}}\Big{(}\frac{\tau_{k}(-\varphi^{\prime}-s)}{A_{s,k}}\Big{)}^{1/n}$
$\displaystyle\geq$
$\displaystyle\frac{n^{2}\varepsilon}{n+1}(-\psi_{s,k}+\Lambda)^{-\frac{1}{n+1}}\frac{(-\varphi^{\prime}-s)^{1/n}}{A^{1/n}_{s,k}}.$

From the choice of the constant $\varepsilon$ in (2.21), the inequality (2.26)
easily yields that $\Psi(x_{0})\leq 0$. Hence we finish the proof the claim.
We then obtain that on $\Omega_{s}$,

(2.27)
${(-\varphi^{\prime}-s)^{\frac{n+1}{n}}}\leq\beta_{1}^{\frac{n+1}{n}}{A_{s,k}^{1/n}}(-\psi_{s,k}+\Lambda).$

We fix a constant $q>n$, and by Lemma 2.3, there exists a constant $C_{1}>0$
such that $\frac{1}{V_{\omega}}\int_{X}(-\psi_{s,k})^{q}\omega^{n}\leq C_{1}$.
Taking $q$-th power on both sides of (2.27) and integrating the resulted
inequality over $\Omega_{s}$ yield that

(2.28)
$\displaystyle\frac{1}{V_{\omega}}\int_{\Omega_{s}}(-\varphi^{\prime}-s)^{q\frac{n+1}{n}}\omega^{n}$
$\displaystyle\leq$ $\displaystyle
C_{2}{A_{s,k}^{q/n}}\frac{1}{V_{\omega}}\int_{X}((-\psi_{s,k})^{q}+\Lambda^{q})\omega^{n}$
$\displaystyle\leq$ $\displaystyle C_{2}(C_{1}+\Lambda^{q})A_{s,k}^{q/n},$

where $C_{2}$ is a constant depending on $q$ and $\beta_{1}$. Letting
$k\to\infty$ in (2.28) we see that this inequality remains true if $A_{s,k}$
is replaced by $A_{s}$ (along with the $A_{s,k}$ in the constant $\Lambda$ in
(2.22)).

We define a monotone function as in [14]

$\phi:{\mathbb{R}}_{\geq 0}\to{\mathbb{R}}_{\geq 0}\quad\mbox{by
}\quad\phi(s)=\frac{1}{V_{\omega}}\int_{\Omega_{s}}\omega^{n}.$

From Hölder inequality, it follows that

$\displaystyle A_{s}$ $\displaystyle=$
$\displaystyle\frac{1}{V_{\omega}}\int_{\Omega_{s}}(-\varphi^{\prime}-s)\omega^{n}$
$\displaystyle\leq$
$\displaystyle\Big{(}\frac{1}{V_{\omega}}\int_{\Omega_{s}}(-\varphi^{\prime}-s)^{\frac{q(n+1)}{n}}\omega^{n}\Big{)}^{\frac{n}{q(n+1)}}\phi(s)^{\frac{q(n+1)-n}{q(n+1)}}$
$\displaystyle\leq$
$\displaystyle(C_{2}(C_{1}+\Lambda^{q})A_{s}^{q/n})^{\frac{n}{q(n+1)}}\phi(s)^{\frac{q(n+1)}{q(n+1)-n}}$
$\displaystyle=$
$\displaystyle(C_{2}(C_{1}+\Lambda^{q}))^{\frac{n}{q(n+1)}}A_{s}^{\frac{1}{n+1}}\phi(s)^{\frac{q(n+1)-n}{q(n+1)}}.$

Hence we have

(2.30)
$A_{s}\leq(C_{2}(C_{1}+\Lambda^{q}))^{\frac{1}{q}}\phi(s)^{1+\frac{1}{n}-\frac{1}{q}}=C_{3}\phi(s)^{1+\frac{1}{n}-\frac{1}{q}},$

where the constant $C_{3}$ is given by
$(C_{2}(C_{1}+\Lambda^{q}))^{\frac{1}{q}}$ and is uniformly bounded since
$A_{s}$ is bounded from Lemma 2.4. From (2.30) we get

(2.31) $r\phi(s+r)\leq C_{3}\phi(s)^{1+\frac{1}{n}-\frac{1}{q}},\quad\forall
r\in(0,1)\mbox{ and }s\geq 0.$

It is a classic fact due to De Giorgi [18], that (2.31) implies
$\Omega_{S_{\infty}}=\emptyset$, for
$S_{\infty}=\frac{1}{1-2^{-nq/(q-n)}}+s_{0},$ where $s_{0}>0$ is a number that
satisfies $\phi(s_{0})\leq(2C_{3})^{(q-n)/nq}$.

We note that

$\phi(s_{0})\leq\frac{1}{s_{0}}\frac{1}{[\omega]^{n}}\int_{X}(-\varphi^{\prime})\omega^{n}\leq\frac{C_{7}}{s_{0}},$

where the last inequality follows from Lemma 2.4 and $C_{7}>0$ is the constant
in that lemma. So we can take $s_{0}=C_{7}(2C_{3})^{-(q-n)/nq}$. Combining the
above, we conclude that

(2.32) $\sup_{X}(-\varphi^{\prime})\leq
S_{\infty}=C_{7}(2C_{3})^{-(q-n)/nq}+\frac{1}{1-2^{-nq/(q-n)}}.$

This completes the proof of Theorem 2.1. Q.E.D.

Remark 3.1. We only require the Kähler metric $\omega$ to be in the set
${\mathcal{W}}$ for the purpose of verifying the two integrability conditions
stated in Lemma 2.3 and Lemma 2.4. It is important to note that the proof of
Theorem 2.1 remains valid when these two lemmas are satisfied. Therefore,
there exist alternative choices for the family of metrics $\omega$ beyond the
set $\mathcal{W}$ that can ensure the validity of these lemmas, and
consequently, Theorem 2.1. For instance, in case $\Gamma=\Gamma_{n}$, we can
impose the following assumptions on $\omega$ and the subsolutions
$\chi^{\prime}$:

(2.33) $\omega\leq
K_{1}\omega_{X},\quad\frac{1}{V_{\omega}}\frac{\omega^{n}}{\omega^{n}_{X}}\leq
K_{2},\quad-\kappa_{1}\omega\leq\chi^{\prime}\leq\kappa_{2}\omega.$

Under these assumptions, Lemma 2.3 holds from the $\alpha$-invariant estimates
readily. Note that
$\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}>0$ and
hence
$\kappa_{2}K_{2}\omega_{X}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}>0$.
Green’s formula for the fixed metric $\omega_{X}$ shows
$\int_{X}(-\varphi^{\prime})\omega_{X}^{n}$ is bounded depending only on the
given parameters. Lemma 2.4 is also true by the second inequality in (2.33).

##  3\. The Hermitian case

In this section, we will prove the $L^{\infty}$ estimates when the metric
$\omega$ is fixed and only Hermitian. Let $X$ be a compact complex manifold
without boundary. As in the last section, we assume that
$\chi^{\prime}=\chi-\sqrt{-1}\partial\overline{\partial}\theta$ is a
$(\delta,R)$-subsolution to (1.1) which satisfies

(3.1) $-n\kappa_{1}\omega\leq\chi^{\prime}\leq\kappa_{2}\omega.$

######  Theorem 3.1.

Let the assumptions be given as above, and $\varphi$ is a $C^{2}$ solution to
the equation (1.1), then there exists a constant $C>0$ depending only on
$n,\omega$, and $\delta,R,\kappa_{1},\kappa_{2}$ such that

(3.2)
${\mathrm{sup}}_{X}|(\varphi+\theta)-{\mathrm{sup}}_{X}(\varphi+\theta)|\leq
C.$

We denote $\varphi^{\prime}=\varphi+\theta$ and assume without loss of
generality that $\sup_{X}\varphi^{\prime}=0$. Since the metric $\omega$ is
fixed, there is a constant $r_{0}=r_{0}(n,\omega,X)>0$ such that for any point
$z_{0}\in X$, there exist holomorphic coordinates $z=(z_{1},\ldots,z_{n})$ at
$z_{0}$ such that on the ball
$B(z_{0},2r_{0})=\\{z:~{}\sum_{j}|z_{j}|^{2}<(2r_{0})^{2}\\}$ the metric
$\omega=g_{i\bar{j}}\sqrt{-1}dz_{i}\wedge d\bar{z}_{j}$ satisfies

(3.3) $\frac{1}{2}\delta_{ij}\leq g_{i\bar{j}}\leq 2\delta_{ij}.$

We assume $x_{0}$ is a minimum point of $\varphi^{\prime}$, i.e.
$\min_{X}\varphi^{\prime}=\varphi^{\prime}(x_{0})$, and pick the coordinates
$z$ at $x_{0}$ such that (3.3) holds. Fix
$s_{0}=\frac{\delta}{10}\kappa_{1}r_{0}^{2}>0$. For any $s\in(0,s_{0})$, we
consider the function $u_{s}$ defined on the ball
$B(x_{0},r_{0})=\\{z~{}:~{}|z|<r_{0}\\}$ as

$u_{s}(z)=-\varphi^{\prime}(x_{0})+\varphi^{\prime}(z)+\frac{\delta}{4}|z|^{2}-s,$

and define the level set of $u_{s}$

$U_{s}:=\\{z\in B(x_{0},r_{0})|~{}u_{s}(z)<0\\}.$

Since $u_{s}|_{\partial B(x_{0},r_{0})}>0$, we see that $U_{s}$ is relatively
compact in $B(x_{0},r_{0})$. We consider the solution to the Dirichlet problem

$(\sqrt{-1}\partial\overline{\partial}\psi_{s,k})^{n}=\frac{\tau_{k}(-u_{s})}{A_{s,k}}\omega^{n},\,\mbox{in
}B(x_{0},r_{0}),$

with $\psi_{s,k}=0$ on $\partial B(x_{0},r_{0})$, where we choose the constant
$A_{s,k}>0$ such that

(3.4)
$\int_{B(x_{0},r_{0})}(\sqrt{-1}\partial\overline{\partial}\psi_{s,k})^{n}=1$

in other words

$A_{s,k}:=\int_{B(x_{0},r_{0})}\tau_{k}(-u_{s})\omega^{n}\to
A_{s}:=\int_{\Omega_{s}}(-u_{s})\omega^{n}.$

Note that the solution $\psi_{s,k}\leq 0$ exists and is unique by [3]. We next
consider the test function

$\Phi=-\varepsilon(-\psi_{s,k})^{\frac{n}{n+1}}-u_{s},\mbox{ in
}B(x_{0},r_{0}),$

where

(3.5)
$\varepsilon=(\frac{n+1}{n})^{\frac{n}{n+1}}(R+\delta+\kappa_{1})^{\frac{n}{n+1}}A_{s,k}^{\frac{1}{n+1}}=:\beta_{3}A_{s,k}^{\frac{1}{n+1}}.$

As in the last section, we claim $\Phi\leq 0$. It suffices to prove this when
the maximum of $\Phi$ is achieved at some point $z_{0}\in U_{s}\subset
B(x_{0},r_{0})$. By maximum principle, we have that at the point $z_{0}$,
$\sqrt{-1}\partial\overline{\partial}\Phi\leq 0$, which implies that at
$z_{0}$

$\displaystyle\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}$
$\displaystyle\geq$
$\displaystyle\frac{n\varepsilon}{n+1}(-\psi_{s,k})^{-\frac{1}{n+1}}\sqrt{-1}\partial\overline{\partial}\psi_{s,k}-\frac{\delta}{4}\omega_{{\mathbf{C}}^{n}}$
$\displaystyle\geq$
$\displaystyle\frac{n\varepsilon}{n+1}(-\psi_{s,k})^{-\frac{1}{n+1}}\sqrt{-1}\partial\overline{\partial}\psi_{s,k}-\frac{\delta}{2}\omega.$

Hence at $z_{0}$ it holds that

(3.6)
$\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}>\frac{n\varepsilon}{n+1}(-\psi_{s,k})^{-\frac{1}{n+1}}\sqrt{-1}\partial\overline{\partial}\psi_{s,k}+\chi^{\prime}-{\delta}\omega.$

This inequality shows that
$\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}=\chi^{\prime}-\delta\omega+\rho$
for some positive definite $(1,1)$-form $\rho$. In terms of eigenvalues, this
means that

$\lambda[h_{\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}}]\in\lambda[h_{\chi^{\prime}}]-\delta{\mathbf{1}}+\Gamma_{n}.$

By the definition of the $(\delta,R)$-subsolution $\chi^{\prime}$, we have
$|\lambda_{i}|\leq R$ for each $i=1,2,\ldots,n$, where
$(\lambda_{1},\ldots,\lambda_{n})$ are the eigenvalues of
$\omega^{-1}\cdot(\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime})|_{z_{0}}$.
Again by (3.6) we have at $z_{0}$

$\displaystyle n(R+\kappa_{1}+\delta)$ $\displaystyle\geq$ $\displaystyle{\rm
tr}_{\omega}(\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}-\chi^{\prime}+\delta\omega)$
$\displaystyle\geq$
$\displaystyle\frac{n\varepsilon}{n+1}(-\psi_{s,k})^{-\frac{1}{n+1}}{\rm
tr}_{\omega}(\sqrt{-1}\partial\overline{\partial}\psi_{s,k})$
$\displaystyle\geq$
$\displaystyle\frac{n^{2}\varepsilon}{n+1}(-\psi_{s,k})^{-\frac{1}{n+1}}\Big{(}\frac{(\sqrt{-1}\partial\overline{\partial}\psi_{s,k})^{n}}{\omega^{n}}\Big{)}^{1/n}$
$\displaystyle\geq$
$\displaystyle\frac{n^{2}\varepsilon}{n+1}(-\psi_{s,k})^{-\frac{1}{n+1}}\frac{(-u_{s})^{1/n}}{A_{s,k}^{1/n}},$

which is equivalent to $\Phi|_{z_{0}}\leq 0$ by the choice of $\varepsilon$ in
(3.5). Therefore, on $U_{s}$ we have

(3.7)
${(-u_{s})^{{(n+1)/}{n}}}{}\leq\beta_{3}^{\frac{n+1}{n}}A_{s,k}^{1/n}(-\psi_{s,k}).$

By the uniform $\alpha$-invariant estimates (see [18]) for $\psi_{s,k}$ which
satisfies (3.4), there exist uniform constants $\alpha=\alpha(n,r_{0},\omega)$
and $C=C(n,r_{0},\omega)$ such that
$\int_{B(x_{0},r_{0})}e^{-\alpha\psi_{s,k}}\omega^{n}\leq C$, hence for any
$q>n$ it holds that

(3.8) $\int_{B(x_{0},r_{0})}(-\psi_{s,k})^{q}\omega^{n}\leq
C_{0}^{\prime}=C_{0}^{\prime}(n,r_{0},\omega,q).$

Taking $q$-th power on both sides of (3.7) and integrating over $U_{s}$, we
have from (3.8) that

(3.9)
$\int_{U_{s}}(-u_{s})^{\frac{q(n+1)}{n}}\omega^{n}\leq\beta_{3}^{\frac{q(n+1)}{n}}A_{s,k}^{\frac{q}{n}}\int_{B(x_{0},r_{0})}(-\psi_{s,k})^{q}\omega^{n}\leq
C_{0}^{\prime}\beta_{3}^{\frac{q(n+1)}{n}}A_{s,k}^{\frac{q}{n}}.$

Letting $k\to\infty$, inequality (3.9) continues to hold when $A_{s,k}$ is
replaced by $A_{s}$. By Hölder inequality, we can argue similarly as in (2) to
derive

(3.10)
$A_{s}\leq(C_{0}^{\prime}\beta_{3}^{\frac{q(n+1)}{n}})^{\frac{1}{q}}\hat{\phi}(s)^{1+\frac{1}{n}-\frac{1}{q}}=:C_{1}^{\prime}\hat{\phi}(s)^{1+\frac{1}{n}-\frac{1}{q}},$

where
$C_{1}^{\prime}=(C_{0}^{\prime}\beta_{3}^{\frac{q(n+1)}{n}})^{\frac{1}{q}}$
and $\hat{\phi}(s)$ is a monotone function given by

$\hat{\phi}(s)=\int_{U_{s}}\omega^{n},\quad\forall s\in(0,s_{0}).$

From (3.10) we obtain

(3.11) $t\hat{\phi}(s-t)\leq
C_{1}^{\prime}\hat{\phi}(s)^{1+\frac{1}{n}-\frac{1}{q}},\quad\forall\,0<t<s<s_{0}.$

Then an iteration argument (see e.g. Lemma 7 in [10]) shows that there exists
a constant $c_{0}>0$ that depends on $n,q,C_{0}^{\prime},$ and $s_{0}$ such
that $\hat{\phi}(s_{0})\geq c_{0}$. We observe that on $U_{s_{0}}$,

$u_{s_{0}}(z)<0\quad\Rightarrow\quad-\varphi^{\prime}(x_{0})\leq-\varphi^{\prime}(z)-\frac{\delta}{4}|z|^{2}+s_{0}.$

Integrating this inequality over $U_{s_{0}}$ yields

(3.12)
$c_{0}(-\varphi^{\prime}(x_{0}))\leq\int_{U_{s_{0}}}(-\varphi^{\prime}(x_{0}))\omega^{n}\leq\int_{B(x_{0},r_{0})}(-\varphi^{\prime})\omega^{n}+C(n,r_{0})s_{0}.$

From (3.12), we see that

$-\varphi^{\prime}(x_{0})=-{\mathrm{inf}}_{X}\varphi^{\prime}\leq
C(\|\varphi^{\prime}\|_{L^{1}(X,\omega^{n})}+1),$

for some constant $C>0$ that depends on $n,\kappa_{1},q$ and $\omega$. To
complete the proof it suffices to derive an $L^{1}$ estimate of
$\varphi^{\prime}$. This follows from the fact that
$\lambda[h_{\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}}]\in\Gamma\subset\Gamma_{1}$,
hence

${\rm tr}_{\omega}\chi^{\prime}+\Delta_{\omega}\varphi^{\prime}\geq 0,$

where $\Delta_{\omega}\varphi^{\prime}={\rm
tr}_{\omega}(\sqrt{-1}\partial\overline{\partial}\varphi^{\prime})$ is the
rough Laplacian of the Hermtian metric $\omega$. Given the assumption in (3.1)
that ${\rm tr}_{\omega}\chi^{\prime}\leq\kappa_{2}$, we have the linear
differential inequality $\Delta_{\omega}\varphi^{\prime}\geq-\kappa_{2}$ on
$X$. By a local covering argument (see e.g. Lemma 8 in [10]) and standard
elliptic estimates [8], we have

$\int_{X}(-\varphi^{\prime})\omega^{n}\leq C,$

for some $C>0$ depending on $n,\omega,\kappa_{2}$. This finishes the proof of
Theorem 3.1.

##  4\. Other forms of subsolutions

In this section, we discuss another form of subsolutions in the case of
$\Gamma=\Gamma_{n}$. To simplify the exposition, we make the convention that
for a vector $\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\Gamma_{n}$, i.e.
$\lambda_{i}>0$ for all $i$, we define
$\tilde{\lambda}=(\frac{1}{\lambda_{1}},\ldots,\frac{1}{\lambda_{n}})\in\Gamma_{n}$.
Abusing notation, we sometimes write
$\tilde{\lambda}_{i}=\frac{1}{\lambda_{i}}$. Associated with the operator $f$
which satisfies the condition (ii) in Section 1 and the condition of
homogeneity of degree one, we define a new fully nonlinear operator
$\tilde{f}:\Gamma_{n}\to{\mathbf{R}}_{>0}$ as

(4.13)
$\tilde{f}(\tilde{\lambda}):=\frac{1}{f({\tilde{\lambda}_{1}}^{-1},\ldots,{\tilde{\lambda}_{n}}^{-1})}=\frac{1}{f(\lambda)}.$

Clearly $\tilde{f}$ is also a homogeneous function of degree one and
$\frac{\partial\tilde{f}}{\partial\tilde{\lambda}_{i}}(\tilde{\lambda})>0$ for
any $\tilde{\lambda}\in\Gamma_{n}$. Furthermore, we impose the extra
conditions on $f$ or $\tilde{f}$ as follows.

(iii): $\tilde{f}$ is concave in $\tilde{\lambda}\in\Gamma_{n}$.

######  Definition 4.1.

We say a metric
$\chi^{\prime}=\chi-\sqrt{-1}\partial\overline{\partial}\theta$ is a
$\delta$-subsolution of (1.1), if

(4.14) $\sum_{j\neq
i}\tilde{\mu}_{j}\frac{\partial\tilde{f}}{\partial\tilde{\lambda}_{j}}(\tilde{\mu})\leq(1-\delta)e^{-F(z)},\quad\forall
z\in X,\,\forall i=1,\ldots,n.$

Here $\mu=(\mu_{1},\ldots,\mu_{n})$ is an unordered vector of eigenvalues of
$\omega^{-1}\cdot\chi^{\prime}$, and conventionally we have defined
$\tilde{\mu}_{j}=\frac{1}{\mu_{j}}$ and
$\tilde{\mu}=(\tilde{\mu}_{1},\ldots,\tilde{\mu}_{n})$.

We observe that if $\chi^{\prime}$ satisfies the equation (1.1), by the
homogeneity of degree one condition, we necessarily have
$\sum_{j=1}^{n}\tilde{\mu}_{j}\frac{\partial\tilde{f}}{\partial\tilde{\lambda}_{j}}(\tilde{\mu})=e^{-F(z)}$.
So a solution is indeed a $\delta$-subsolution for some $\delta>0$.

Given a $\delta$-subsolution $\chi^{\prime}$ as in Definition 4.1, we have the
following uniform $L^{\infty}$ estimates for solutions $\varphi$ to (1.1).

######  Theorem 4.1.

Assume the operator $f$ satisfies the additional condition (iii). Suppose
$\varphi$ is a $C^{2}$ solution to the equation (1.1),
$\omega\in{\mathcal{W}}$ is a Kähler metric, and $\chi^{\prime}$ is a
$\delta$-subsolution as in Definition 4.1 such that
$\chi^{\prime}\leq\kappa_{2}\omega$ for some constant $\kappa_{2}>0$. Then
there exists a constant $C>0$ that depends on $\kappa_{2},\delta$ and the
parameters in ${\mathcal{W}}$ such that

${\mathrm{sup}}_{X}|(\varphi+\theta)-{\mathrm{sup}}_{X}(\varphi+\theta)|\leq
C.$

Proof of Theorem 4.1. We follow the notations as in Section 2, and denote
$\varphi^{\prime}=\varphi+\theta$. Let $\psi_{s,k}$ and $\Psi$ be as in (2.18)
and (2.20), respectively. Here the two constants $\varphi$ and $\Lambda$ in
(2.20) are chosen as

$\varepsilon=\Big{(}\frac{n+1}{n}R\Big{)}^{\frac{n}{n+1}}A_{s,k}^{\frac{1}{n+1}},\quad\Lambda=\Big{(}\frac{2n\kappa_{2}\varepsilon}{(n+1)\delta}\Big{)}^{n+1},$

and here $R:=\frac{(2-\delta)(1-\delta)}{\delta}\kappa_{2}>0$. With these
choice of constants, we claim $\Psi\leq 0$. At a maximum point
$x_{0}\in\Omega_{s}$ of $\Psi$, an analogous inequality as (2.23) holds, which
gives

(4.15)
$\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime}>(1-0.5\delta)\chi^{\prime}.$

We take normal coordinates of $\omega$ at $x_{0}$, and consider the Hermitian
matrices
$B^{\prime}=(\chi^{\prime}+\sqrt{-1}\partial\overline{\partial}\varphi^{\prime})_{i\bar{j}}|_{x_{0}}$
and $B=(\chi^{\prime})_{i\bar{j}}|_{x_{0}}$. $B^{\prime}$ and $B$ may not be
necessarily diagonal under these coordinates. We may assume there eigenvalues
are listed in an increasing order, i.e.

$0<\lambda_{1}\leq\cdots\leq\lambda_{n},\quad\mbox{and}\quad
0<\mu_{1}\leq\cdots\leq\mu_{n}.$

The inequality (4.15) means $B^{\prime}>(1-\delta)B.$ It follows from an
elementary fact from linear algebra (see Lemma 4.1 below) that

(4.16) $\lambda_{i}\geq\mu_{i}(1-0.5\delta),\quad\forall\,i=1,\ldots,n.$

By the condition (i’) that $\tilde{f}(\tilde{\lambda})$ is concave in
$\tilde{\lambda}$, it follows that

(4.17)
$\tilde{f}(\tilde{\lambda})\leq\tilde{f}(\tilde{\mu})+\sum_{i=1}^{n}\frac{\partial\tilde{f}}{\partial\tilde{\lambda}_{i}}(\tilde{\mu})\cdot(\tilde{\lambda}_{i}-\tilde{\mu}_{i})=\sum_{i=1}^{n}\frac{\partial\tilde{f}}{\partial\tilde{\lambda}_{i}}(\tilde{\mu})\cdot\tilde{\lambda}_{i},$

where we have used the condition that $\tilde{f}(\tilde{\lambda})$ is
homogeneous of degree one. Here we need the subsolution condition (4.14) for
$\chi^{\prime}$. For each fixed $i=1,\ldots,n$, we have from (4.17) that

(4.18)
$e^{-F(x_{0})}\leq\frac{\partial\tilde{f}}{\partial\tilde{\lambda}_{i}}(\tilde{\mu})\cdot\tilde{\lambda}_{i}+\sum_{j\neq
i}\frac{\partial\tilde{f}}{\partial\tilde{\lambda}_{j}}(\tilde{\mu})\cdot\tilde{\lambda}_{j}\leq\frac{\partial\tilde{f}}{\partial\tilde{\lambda}_{i}}(\tilde{\mu})\cdot\tilde{\lambda}_{i}+\frac{1-\delta}{1-0.5\delta}e^{-F(x_{0})},$

which implies that

$\frac{\delta}{2-\delta}e^{-F(x_{0})}\leq\frac{\partial\tilde{f}}{\partial\tilde{\lambda}_{i}}(\tilde{\mu})\cdot\tilde{\lambda}_{i}\leq\frac{\tilde{\lambda}_{i}}{\tilde{\mu}_{i}}(1-\delta)e^{-F(x_{0})}.$

Hence

$\lambda_{i}\leq\frac{(2-\delta)(1-\delta)}{\delta}\mu_{i}\leq\frac{(2-\delta)(1-\delta)}{\delta}\kappa_{2}=R.$

The remaining arguments are the same as those in Section 2. The proof of
Theorem 4.1 is completed. Q.E.D.

######  Lemma 4.1.

Suppose $A,B$ are both $n\times n$ Hermitian matrices satisfying $B\geq A$,
i.e. $B-A$ is semi-positive definite. If we denote $\lambda_{i}(A)$ the
eigenvalues of $A$ in the increasing order, i.e.
$0<\lambda_{1}(A)\leq\cdots\leq\lambda_{n}(A)$, and similar notation for
$\lambda_{i}(B)$, then

$\lambda_{i}(B)\geq\lambda_{i}(A),\quad\forall i=1,\ldots,n.$

Proof of Lemma 4.1. It follows from the Courant-Fischer min-max Theorem that

$\lambda_{i}(A)=\min\Big{\\{}\max\\{\langle Az,z\rangle:~{}z\in
U,\,|z|=1\\},~{}U\mbox{ with dim}_{{\mathbf{C}}}U=i\Big{\\}},$

where the minimum is taken over all complex vector spaces
$U\subset{\mathbf{C}}^{n}$ with dimension $i$. For each fixed vector space $U$
we have

$\max\\{\langle Bz,z\rangle:~{}z\in U,\,|z|=1\\}\geq\max\\{\langle
Az,z\rangle:~{}z\in U,\,|z|=1\\}.$

Taking minimum over such $U$’s gives the desired inequality. Q.E.D.

###  4.1. Examples

####  4.1.1. The $J$-equation

We consider the following form of $J$-equation:

(4.19)
$(\chi+\sqrt{-1}\partial\overline{\partial}\varphi)^{n}=ne^{F}\omega\wedge(\chi+\sqrt{-1}\partial\overline{\partial}\varphi)^{n-1},\quad\sup_{X}\varphi=0.$

In the notation of $f(\lambda[h_{\varphi}])$, this equation is equivalent to

$f(\lambda)=\frac{\sigma_{n}(\lambda)}{\sigma_{n-1}(\lambda)}=e^{F},$

where $\lambda\in\Gamma_{n}$ is the vector of eigenvalues of
$\omega^{-1}\cdot(\chi+\sqrt{-1}\partial\overline{\partial}\varphi)$.
Rewriting the nonlinear operator $f$ in terms of $\tilde{f}$, we see that
$\tilde{f}(\tilde{\lambda})=\sigma_{1}(\tilde{\lambda})$, which satisfies the
required conditions. The subsolution condition (4.14) for $\chi^{\prime}$ with
this equation states that

$\sum_{j\neq
i}\frac{1}{\mu_{j}}\leq(1-\delta){e}^{-F(z)},\quad\forall\,i=1,\ldots,n,$

which is just the subsolution condition introduced in [22] (see also [23]). We
mention that for the $J$-equation, it is proved in [21] that the existence of
a subsolution is equivalent to a Nakai-Moishezon criterion, thus confirming a
conjecture of [19].

####  4.1.2. Hessian type equations

The following equation is studied in [7], when $1\leq k\leq n$

$(\chi+\sqrt{-1}\partial\overline{\partial}\varphi)^{n}=e^{F}\omega^{n-k}\wedge(\chi+\sqrt{-1}\partial\overline{\partial}\varphi)^{k},\quad\sup_{X}\varphi=0.$

The subsolution condition (4.14) for $\chi^{\prime}$ with this equation is the
one studied in [7].

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