Let $(X,\omega)$ be a compact Kähler manifold, $\chi$ a closed and non-negative $(1,1)$-form, and set

$\displaystyle\omega_{t}=\chi+t\omega,\qquad t\in(0,1].$

(2.1)

Let $f,h\in C^{\infty}$ are smooth functions normalized by

$$\int_{X}e^{f}\omega^{n}=\int_{X}e^{h}\omega^{n}=\int_{X}\omega^{n}=1,$$

and consider the following complex Hessian equations

$$(\omega_{t}+i\partial\bar{\partial}u_{t})^{k}\wedge\omega^{n-k}=c_{t}e^{f}\omega^{n},\quad(\omega_{t}+i\partial\bar{\partial}v_{t})^{k}\wedge\omega^{n-k}=c_{t}e^{h}\omega^{n}$$

(2.2)

with the constants $c_{t}$ given by $c_{t}=\int_{X}\omega_{t}^{k}\wedge\omega^{n-k}$. We normalize $u_{t}$ and $v_{t}$ so that

$$\max_{X}(u_{t}-v_{t})=\max_{X}(v_{t}-u_{t}).$$

We will consider three cases:

		$\displaystyle{\mathrm{\mathbf{I}}}:$	$\displaystyle k=n,\mbox{ and }t\in(0,1]$
		$\displaystyle{\mathrm{\mathbf{II}}}:$	$\displaystyle 1\leq k<n,\mbox{ and }\chi=0,\,t=1.$
		$\displaystyle{\mathrm{\mathbf{III}}}:$	$\displaystyle 1\leq k<n,\mbox{ and }t\in(0,1],\,\chi\mbox{ is big i.e. $\int_{X}\chi^{n}>0$}$

which correspond respectively to the Monge-Ampère equations with degenerations, the $k$-th Hessian equation with a fixed background metric, and the $k$-th Hessian equation with degenerations. Different cases correspond to different choices of test functions, and constants. So we will treat the cases separately, when necessary.

For each case, we will make the following assumptions and choice of constants,

		$\displaystyle{\mathrm{\mathbf{I}}}:$	$\displaystyle\|e^{h}\|_{L^{1}(\,{\rm log}\,L)^{p_{1}}(\omega^{n})},\|e^{f}\|_{L^{1}(\,{\rm log}\,L)^{p_{1}}(\omega^{n})}\leq K,\,p_{1}>n$
		$\displaystyle{\mathrm{\mathbf{II}}}:$	$\displaystyle\|e^{h}\|_{L^{p_{2}}(\omega^{n})},\|e^{f}\|_{L^{p_{2}}(\omega^{n})}\leq K,\,p_{2}>\frac{n}{k},\mbox{ and }q_{2}=\frac{p_{2}-1}{p_{2}-n/k}$
		$\displaystyle{\mathrm{\mathbf{III}}}:$	$\displaystyle\|e^{h}\|_{L^{p_{3}}(\omega^{n})},\|e^{f}\|_{L^{p_{3}}(\omega^{n})}\leq K,\,p_{3}>\frac{n^{2}}{k},\mbox{ and }q_{3}=\frac{p_{3}-1}{p_{3}-n/k}$

for a fixed constant $K>0$.

The equations (2.2) admit smooth solutions by [15, 5]. By [9] (also [12, 4]), under the above assumptions, the oscillations ${\mathrm{osc}}u_{t}$ and ${\mathrm{osc}}v_{t}$ of the solutions are uniformly bounded independently of $t$ in all three cases ${\mathrm{\mathbf{I}}},{\mathrm{\mathbf{I}I}}$, and ${\mathrm{\mathbf{I}II}}$. Let $\beta_{0}>1$ denote such an upper bound depending only on $n,k,\omega,\chi,p_{a}$ and $10K$.

To start with, we will define a positive function $\gamma_{a}(r)$ with $\gamma_{a}(r)\to 0$ as $r\to 0$. Each case has different choice of such a function. We define $\gamma_{a}$ case by case:

		$\displaystyle{\mathrm{\mathbf{I}}}:$	$\displaystyle\gamma_{1}(r)=r^{\frac{1}{\delta_{1}}-n+1},\mbox{ where }\delta_{1}=\frac{p_{1}-n}{p_{1}n}<\frac{1}{n}$
		$\displaystyle{\mathrm{\mathbf{II}}}:$	$\displaystyle\gamma_{2}(r)=r^{(n+1)q_{2}-n}$
		$\displaystyle{\mathrm{\mathbf{III}}}:$	$\displaystyle\gamma_{3}(r)=r^{(n+1)q_{3}-n}$

We observe that this theorem improves on all results known so far. More specifically in case I, we get uniform stability for a degenerating family, and it does not even matter whether $\chi$ is big or not. Kolodziej [12] proved this case for a fixed Kähler metric, and Dinew and Zhang [6] proved it for a fixed big class. In case II, we slightly sharpen the known stability result in [4], where the RHS in the inequality is $\|e^{f}-e^{h}\|_{L^{q^{\prime}}}$ for some $q^{\prime}>1$, while we are able to prove the inequality for $q^{\prime}=1$. In case III, we obtain a uniform stability theorem for Hessian equations when the class remains big. This is completely new, and relies in particular on the uniform $L^{\infty}$ estimate in [9] for solutions with degenerating big classes.

We note that the exponent $1/(n+3+\sigma_{a})$ is not sharp in general. The sharp exponent can be obtained by replacing Lemma 1 below by a result from [6]. We leave the details to the interested readers.