Stability estimates for a non-linear partial differential equation are estimates for how much the solution can vary, given the size of the variation of the right hand side. Clearly, they are of great theoretical as well as practical importance. Such estimates had been obtained by Kolodziej [12] for the complex Monge-Ampère equation, by Dinew and Kolodziej [4] for complex Hessian equations, and by Dinew and Zhang [6] for Monge-Ampère equations when the background metric is not necessarily Kähler, but just big. In all cases, the proofs made extensive use of pluripotential theory and the background metric was fixed. It remained an open question whether these estimates can be established without pluripotential theory, and whether they can be made uniform under degenerations of the background metric, a situation which arises frequently in geometric applications.

In [9], the authors developed a method for obtaining sharp $L^{\infty}$ estimates for the complex Monge-Ampère equation without pluripotential theory. As explained in greater detail there, the method of [9] built on works of Wang, Wang, Zhou [14] and of Chen and Cheng [3], particularly on the last two authors’ idea of considering an associated complex Monge-Ampère equation. It achieved the stated goal of giving an alternate PDE proof of $L^{\infty}$ estimates for the complex Monge-Ampère equation, but it also went considerably beyond in, on one hand, applying to more general fully non-linear equations, and on the other hand, allowing the background metrics to degenerate. It can also give sharp gradient estimates [10], improving on the estimates in e.g. [13, 2, 8, 7, 11].

The main goal of the present paper is to obtain stability estimates for the complex Monge-Ampère and Hessian equations which are uniform under degenerations. We use the method of [9]. We recover in the process the stability estimates of [12, 4, 6], this time without pluripotential theory. Our estimates are also uniform under general degenerations of the background metric in the case of the Monge-Ampère equation, and under degenerations to a big class in the case of Hessian equations. Thus we answer in the positive both questions asked above.

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.

For the proof, it is convenient to restate the theorem as follows. Under the above assumptions, if in each case we have

$$\|e^{f}-e^{h}\|_{L^{1}(\omega^{n})}\leq\gamma_{a}(r)r^{n+3},$$

(3.1)

then there is a small $r_{0}>0$ such that for all $0<r\leq r_{0}$

$$\sup_{X}|u_{t}-v_{t}|\leq Cr,$$

for all $t$ listed in each case in (2) and $C$ depends only on $n,k,\omega,\chi$, $K>0$ and the corresponding $p_{a}$ in each case. This is the version which we shall prove.

We begin with a lemma due to Kolodziej [12]. The proof is almost identical to that in [12], but since we would like to avoid the use of pluripotential theory, some additional smoothing is needed in the proof, and we provide a full proof of this lemma.

Choose a small $\bar{r}_{0}\in(0,\frac{1}{10})$ such that $\gamma_{a}(\bar{r}_{0})\bar{r}_{0}^{n}\leq\frac{1}{5}$. We then fix an $0<r<\bar{r}_{0}$. We remark that all relevant constants are independent of $r$. Later on we will choose an even smaller $r_{0}>0$.

By switching the roles of $u_{t}$ and $v_{t}$ if necessary, we may assume

$$\int_{\{v_{t}\leq u_{t}\}}(e^{f}+e^{h})\omega^{n}\leq 1.$$

Denote $E_{j}:=\{v_{t}\leq u_{t}-j\beta_{0}r\}$. The next lemma states that over the set $E_{2}$, the integral of $e^{h}$ is small.

$\bullet$ in case I, we have $\|e^{h_{j}}\|_{L^{1}(\,{\rm log}\,L)^{p_{1}}}\leq 5K$,

$\bullet$ in case II, we have $\|e^{h_{j}}\|_{L^{p_{2}}}\leq 5K$,

$\bullet$ in case III, we have $\|e^{h_{j}}\|_{L^{p_{3}}}\leq 5K$.

We solve the following Hessian equations which admit known to admit unique smooth solutions [15, 5],

$$(\omega_{t}+i\partial\bar{\partial}\rho_{j})^{k}\wedge\omega^{n-k}=c_{t}e^{h_{j}}\omega^{n},\quad\sup_{X}\rho_{j}=0\mbox{ and }\omega_{t}+i\partial\bar{\partial}\rho_{j}\in\Gamma_{k},$$

where $\Gamma_{k}$ is the usual open convex cone in $k$-th Hessian equations. By the choice of $\beta_{0}$, we have $-\beta_{0}\leq\rho_{j}\leq 0$ (see [9]). The following Newton inequality holds pointwise for any $1\leq l\leq k$

$$\omega_{t,u_{t}}^{l}\wedge\omega_{t,\rho_{j}}^{k-l}\wedge\omega^{n-k}\geq\Big(\frac{\omega_{t,u_{t}}^{k}\wedge\omega^{n-k}}{\omega^{n}}\Big)^{l/k}\Big(\frac{\omega^{k}_{t,\rho_{j}}\wedge\omega^{n-k}}{\omega^{n}}\Big)^{(k-l)/k}\omega^{n}.$$

Then on the set $E_{0}\backslash G$ where $G=\{e^{f}\leq(1-r^{2})e^{h}\}$, we have

$$\omega_{t,u_{t}}^{l}\wedge\omega_{t,\rho_{j}}^{k-l}\wedge\omega^{n-k}\geq c_{t}(1-r^{2})^{l/k}(\frac{3}{2})^{(k-l)/k}e^{h}\omega^{n}.$$

It follows that on $E_{0}\backslash G$

	$\displaystyle\omega_{t,r\rho_{j}+(1-r)u_{t}}^{k}\wedge\omega^{n-k}$	$\displaystyle=$	$\displaystyle\sum_{l=0}^{k}\frac{k!}{l!(k-l)!}r^{k-l}(1-r)^{l}\omega_{t,u_{t}}^{l}\wedge\omega_{t,\rho_{j}}^{k-l}\wedge\omega^{n-k}$		(3.3)
		$\displaystyle\geq$	$\displaystyle c_{t}\sum_{l=0}^{k}\frac{k!}{l!(k-l)!}r^{k-l}(1-r)^{l}(1-r^{2})^{l/k}(\frac{3}{2})^{(k-l)/k}e^{h}\omega^{n}$
		$\displaystyle=$	$\displaystyle c_{t}\big((1-r)(1-r^{2})^{1/k}+r(\frac{3}{2})^{1/k}\big)^{k}e^{h}\omega^{n}\geq c_{t}(1+b_{0}r)e^{h}\omega^{n}$

where $b_{0}=\frac{1}{2}((\frac{3}{2})^{1/k}-1)>0$, since $r$ is chosen to be small.

Note that by (3.1)

$$r^{2}\int_{G}e^{h}\omega^{n}\leq\int_{G}(e^{h}-e^{f})\omega^{n}\leq\gamma_{a}(r)r^{n+3},$$

which implies

$$\int_{G}e^{h}\omega^{n}\leq\gamma_{a}(r)r^{n+1}.$$

(3.4)

Adding the same constant to $u_{t}$ and $v_{t}$, we may assume without loss of generality $-\beta_{0}\leq u_{t}\leq 0$. The following inclusion relation holds from the definition

$$E_{2}\subset E:=\{v_{t}\leq r\rho_{j}+(1-r)u_{t}-\beta_{0}r\}\subset E_{0}.$$

All functions involved are smooth so by the comparison principle and (3.3),

	$\displaystyle c_{t}(1+b_{0}r)\int_{E\backslash G}e^{h}\omega^{n}$	$\displaystyle\leq$	$\displaystyle\int_{E}\omega_{t,r\rho_{j}+(1-r)u_{t}}^{k}\wedge\omega^{n-k}$
		$\displaystyle\leq$	$\displaystyle\int_{E}\omega_{v_{t}}^{k}\wedge\omega^{n-k}=c_{t}\int_{E\backslash G}e^{h}\omega^{n}+c_{t}\int_{G}e^{h}\omega^{n}$

Combined with (3.4) this implies

$$\int_{E\backslash G}e^{h}\omega^{n}\leq\frac{1}{b_{0}}\gamma_{a}(r)r^{n}$$

It follows that

$$\int_{E_{2}}e^{h}\omega^{n}\leq\int_{E\backslash G}e^{h}\omega^{n}+\int_{G}e^{h}\omega^{n}\leq(1+\frac{1}{b_{0}})\gamma_{a}(r)r^{n}.$$

The Lemma is proved.

We now come to the proof of Theorem 1 proper. We normalize $u_{t}$ as in the proof of Lemma 1. For $s\geq 0$, we set $\Omega_{s}=\{v_{t}\leq(1-r)u_{t}-3\beta_{0}r-s\}$. Note that $\Omega_{s}\subset E_{2}$ for any $s\geq 0$.

We follow the same strategy as in [9]. We choose a sequence of smooth positive functions $\eta_{j}:{{\bf R}}\to{{{\bf R}}}_{+}$ such that

$$\eta_{j}(x)=x+\frac{1}{j},\quad\mbox{ when }x\geq 0,$$

and

$$\eta_{j}(x)=\frac{1}{2j},\quad\mbox{ when }x\leq-\frac{1}{j},$$

and $\eta_{j}(x)$ lies between $1/2j$ and $1/j$ for $x\in[-1/j,0]$. Clearly $\eta_{j}\to\eta_{\infty}(x)=x\cdot\chi_{{\bf R}_{+}}(x)$ pointwise as $j\to\infty$.

We solve the complex Monge-Ampère equations

$$(\omega_{t}+i\partial\bar{\partial}\psi_{j})^{n}=c_{t}^{{n/}{k}}\frac{\eta_{j}(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)}{A_{s,j}}e^{\frac{n}{k}h}\omega^{n},\quad\sup\psi_{j}=0.$$

As $j\to\infty$ we have by the dominated convergence theorem

$$A_{s,j}=\frac{c_{t}^{\frac{n}{k}}}{V_{t}}\int_{X}\big(\eta_{j}(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)\big)e^{\frac{n}{k}h}\omega^{n}\to A_{s}$$

where the constant $A_{s}$ is defined by

$$A_{s}:=\frac{c_{t}^{\frac{n}{k}}}{V_{t}}\int_{\Omega_{s}}(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)e^{\frac{n}{k}h}\omega^{n}.$$

Consider $\Phi=-\varepsilon\big(-\psi_{j}+\Lambda\big)^{\frac{n}{n+1}}+\big(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s\big)$ where

$$\varepsilon=\Big(\frac{k(n+1)}{n^{2}}\Big)^{\frac{n}{n+1}}c(n,k)^{-\frac{1}{n+1}}A_{s,j}^{\frac{1}{n+1}},\quad\mbox{where $c(n,k)$ is the one in (\ref{eqn:str})}$$

$$\Lambda=\Big(\frac{n}{n+1}\frac{\varepsilon}{r}\Big)^{n+1}=C(n,k)\frac{A_{s,j}}{r^{n+1}}$$

Suppose $\sup\Phi=\Phi(x_{0})$ for some point $x_{0}$ in $X$. If $x_{0}\not\in\Omega_{s}^{\circ}$, then by definition $\Phi(x_{0})<0$. Otherwise $x_{0}\in\Omega_{s}^{\circ}$. We calculate as in [9]. First note that for $G^{i\bar{j}}=\frac{\partial}{\partial(\omega_{t,v_{t}})_{i\bar{j}}}\,{\rm log}\,\sigma_{k}(\omega_{t,v_{t}})$

$${\rm det}G^{i\bar{j}}\geq c(n,k)c_{t}^{-\frac{n}{k}}e^{-\frac{n}{k}h}$$

(3.5)

for some computable constant $c(n,k)>0$. It follows that at $x_{0}$

	$\displaystyle 0$	$\displaystyle\geq$	$\displaystyle G^{i\bar{j}}(\Phi)_{i\bar{j}}(x_{0})$
		$\displaystyle\geq$	$\displaystyle\frac{n^{2}\varepsilon}{n+\delta}(-\psi_{j}+\Lambda)^{-\frac{1}{n+1}}\Big({\rm det}G\cdot{\rm det}\omega_{t,\psi_{j}}\Big)^{1/n}-k+\Big(r-\frac{n\varepsilon}{n+1}\Lambda^{-\frac{1}{n+1}}\Big)G^{i\bar{j}}(\omega_{t})_{i\bar{j}}$
		$\displaystyle\geq$	$\displaystyle\frac{n^{2}\varepsilon}{n+1}(-\psi_{j}+\Lambda)^{-\frac{1}{n+1}}c(n,k)^{1/n}\Big(\frac{-v_{t}+(1-r)u_{t}-3\beta_{0}r-s}{A_{s,j}}\Big)^{1/n}-k.$

By the choice of $\varepsilon$ and $\Lambda$, we deduce that $\Phi(x_{0})\leq 0$. Thus $\Phi\leq 0$ on $X$ and this implies

	$\displaystyle\int_{\Omega_{s}}\,{\rm exp}\,\Big\{c_{0}\Big(\frac{-v_{t}+(1-r)u_{t}-3\beta_{0}r-s}{A_{s,j}^{1/(n+1)}}\Big)^{\frac{n+1}{n}}\Big\}\omega^{n}$	$\displaystyle\leq$	$\displaystyle\int_{\Omega_{s}}\,{\rm exp}\,\Big(-\alpha_{0}\psi_{j}+\alpha_{0}\Lambda\Big)\omega^{n}$		(3.6)
		$\displaystyle\leq$	$\displaystyle C\,{\rm exp}\,\Big\{\frac{A_{s,j}}{r^{n+1}}\Big\}$

for some small $c_{0}=c_{0}(n,k,\omega,\chi)>0$, $C=C(n,k,\omega,\chi)>0$, and $\alpha_{0}$ a fixed number satisfying $0<\alpha_{0}<\alpha(X,\omega)$, where $\alpha(X,\omega)$ is the $\alpha$-invariant of $(X,\omega)$. Letting $j\to\infty$ gives

$$\int_{\Omega_{s}}\,{\rm exp}\,\Big\{c_{0}\Big(\frac{(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)}{A_{s}^{1/(n+1)}}\Big)^{\frac{n+1}{n}}\Big\}\omega^{n}\leq C\,{\rm exp}\,\Big(\frac{A_{s}}{r^{n+1}}\Big).$$

(3.7)

To estimate $\frac{A_{s}}{r^{n+1}}$ in (3.7), we need to consider separately the three cases I, II and III.

So for all cases I, II and III, we get from (3.7) that

$$\int_{\Omega_{s}}\,{\rm exp}\,\Big\{c_{0}\Big(\frac{-v_{t}+(1-r)u_{t}-3\beta_{0}r-s}{A_{s}^{1/(n+1)}}\Big)^{\frac{n+1}{n}}\Big\}\omega^{n}\leq C,$$

(3.8)

for some constant $C>0$ depending on $n,k,\chi,\omega,K$ and the exponents $p_{1},p_{2},p_{3}$ in each case, respectively. In particular this $C$ is independent of the choice of $r\in(0,\bar{r}_{0})$.

We choose $p>n$ as $p=p_{1}$ in case I, and arbitrary and large $p>n$ in cases II and III.

Define $\eta:{{\bf R}}_{+}\to{{\bf R}}_{+}$ by $\eta(x)=(\,{\rm log}\,(1+x))^{p}$. Note that $\eta$ is a strictly increasing function with $\eta(0)=0$, and let $\eta^{-1}$ be its inverse function. If we let

$\displaystyle\Psi:=\frac{c_{0}}{2}\Big(\frac{-v_{t}+(1-r)u_{t}-3\beta_{0}r-s}{A_{s}^{1/(n+1)}}\Big)^{\frac{n+1}{n}}$

(3.9)

then we have for any $z\in\Omega_{s}$, by the generalized Young’s inequality with respect to $\eta$,

	$\displaystyle\Psi(z)^{p}e^{\frac{n}{k}h(z)}$	$\displaystyle\leq$	$\displaystyle\int_{0}^{\,{\rm exp}\,({\frac{n}{k}h(z)})}\eta(x)dx+\int_{0}^{\Psi(z)^{p}}\eta^{-1}(y)dy$
		$\displaystyle\leq$	$\displaystyle e^{\frac{n}{k}h(z)}(1+|h(z)|)^{p}+C(p)e^{2\Psi(z)}$

We integrate both sides in the inequality above over $z\in\Omega_{s}$, and get by (3.8) that

$\displaystyle\int_{\Omega_{s}}\Psi(z)^{p}e^{\frac{n}{k}h(z)}\omega^{n}\leq\|e^{h}\|_{L^{n/k}(\,{\rm log}\,L)^{p}}+C,$

where the constant $C>0$ depends only on $n,k,\omega,\chi,p,K$. In view of the definition of $\Psi$, this implies

$$\int_{\Omega_{s}}(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)^{\frac{(n+1)p}{n}}e^{\frac{n}{k}h}\omega^{n}\leq CA_{s}^{\frac{p}{n}}\big(\|e^{h}\|_{L^{n/k}(\,{\rm log}\,L)^{p}}+1\big).$$

(3.10)

It follows from the Hölder inequality that

	$\displaystyle A_{s}$	$\displaystyle=$	$\displaystyle\frac{c_{t}^{n/k}}{V_{t}}\int_{\Omega_{s}}(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)e^{\frac{n}{k}h}\omega_{X}^{n}$
		$\displaystyle\leq$	$\displaystyle\Big(\frac{c_{t}^{n/k}}{V_{t}}\int_{\Omega_{s}}(-v_{t}+(1-r)u_{t}-3\beta_{0}r-s)^{\frac{(n+1)p}{n}}e^{\frac{n}{k}h}\omega^{n}\Big)^{\frac{n}{(n+1)p}}\cdot\Big(\frac{c_{t}^{n/k}}{V_{t}}\int_{\Omega_{s}}e^{\frac{n}{k}h}\omega^{n}\Big)^{1/q}$
		$\displaystyle\leq$	$\displaystyle CA_{s}^{\frac{1}{n+1}}\Big(\|e^{h}\|_{L^{n/k}(\,{\rm log}\,L)^{p}}+1)\Big)^{\frac{n}{(n+1)p}}\cdot\Big(\frac{c_{t}^{n/k}}{V_{t}}\int_{\Omega_{s}}e^{\frac{n}{k}h}\omega^{n}\Big)^{1/q}$

where $q>1$ satisfies $\frac{n}{p(n+1)}+\frac{1}{q}=1$, i.e. $q=\frac{p(n+1)}{p(n+1)-n}$. The inequality above yields

$$A_{s}\leq C\Big(\|e^{h}\|_{L^{n/k}(\,{\rm log}\,L)^{p}}+1\Big)^{1/p}\cdot\Big(\frac{c_{t}^{n/k}}{V_{t}}\int_{\Omega_{s}}e^{\frac{n}{k}h}\omega^{n}\Big)^{\frac{1+n}{qn}}=B_{0}\Big(\frac{c_{t}^{n/k}}{V_{t}}\int_{\Omega_{s}}e^{\frac{n}{k}h}\omega^{n}\Big)^{1+\delta_{0}}.$$

(3.11)

Observe that the exponent of the integral on the right hand of (3.11) satisfies

$$\frac{1+n}{qn}=\frac{pn+p-n}{pn}=1+\delta_{0}>1,\quad\mbox{for $\delta_{0}:=\frac{p-n}{pn}>0$.}$$

We remark that $\delta_{0}$ can be chosen to be close to $1/n$ in cases II and III by picking $p$ large enough. Furthermore, we note that

$$B_{0}:=C\Big(\|e^{h}\|_{L^{n/k}(\,{\rm log}\,L)^{p}}+1\Big)^{1/p}$$

(3.12)

is a constant depending only on $n,k,\omega,\chi,K$, and the exponents $p_{1},p_{2},p_{3}$ in each case, respectively, and in particular, it is independent of $r$ with $r\in(0,\bar{r}_{0})$.

If we define

$$\phi(s)=\frac{c_{t}^{n/k}}{V_{t}}\int_{\Omega_{s}}e^{\frac{n}{k}h}\omega^{n},$$

then (3.11) shows that if $\Omega_{s+s^{\prime}}\not=\emptyset$ then

$$s^{\prime}\phi(s+s^{\prime})\leq B_{0}\phi(s)^{1+\delta_{0}},\quad\mbox{for all }s^{\prime}\geq 0\mbox{ and }s\geq 0.$$

(3.13)

We now choose $r_{0}<\bar{r}_{0}$ small in each case as follows.