It follows from [K1] that $||u||_{\infty}$ depends on $p$, $M$ and $||f||_{p}$. We can assume that $1<u.$ Choose a finite number of coordinate balls $B^{\prime\prime}_{j}=B(a_{j},3r)$ such that $B^{\prime}_{j}=B(a_{j},r)$ cover $M$ and denote by $B_{j}$ the balls $B(a_{j},2r)$. Since the transition functions for those charts have bounded Jacobians one can find a constant $C>0$ depending only on $M$ and such that for all $\delta<r/2C$ and $z\in B_{j}\cap B_{k}$ we have

where $B_{j}(z,\delta)$ denotes the ball centered at $z$ of radius $\delta$ in the chart $B^{\prime\prime}_{j}.$

Fix non positive functions $\rho_{j}\in C^{\infty}(B_{j})$ with $\rho_{j}=0$ on $B^{\prime}_{j}$, $-1\leq\rho_{j}\leq 0$ in $B_{j}$, and $\rho_{j}=-1$ on a neighbourhood of $\partial B_{j}.$ For some $c_{1}$ depending on $M$ we have

For fixed $\epsilon>0$ we choose $N>2C$ and $\alpha<\frac{1}{q(n+3+\epsilon)+1}$ (with $p,q$ conjugate) satisfying

It is possible since we can choose $N$ so big that $\frac{\log N}{\log C}$ is bigger than two times the left hand side, and then we take $\alpha$ small enough. Using the coordinates in $B^{\prime\prime}_{j}$ we define regularizations

Let us yet define two auxiliary functions

and

By 2.1 we have

We shall approximate $u$ by $\omega$-psh functions $u_{\delta}$ which are created by gluing together the local regularizations $u_{j,\delta}$ (comp. [D]). The function $\eta$ defined above measures the correction term when we pass from local to global regularization. Note that, by continuity of $u$,

Set

It is continuous on $M$ since, by 2.3, the maximum on the right hand side must be attained for $j$ such that $z\in B^{\prime}_{j}.$ Note also that since for $c_{1}\eta(\delta)<1$

one obtains, via an inequality from [BT1] estimating $dd^{c}\max(u,v)$ from below,

if $\delta$ is sufficiently small. To finish the proof we need to verify the following claim.

Claim. $\chi$ is bounded on some nonempty interval $(0,\tilde{\delta}).$

Suppose that $\chi(\delta)>\max(9,\chi(N\delta))$ and $N\delta<r/2.$ Then the set

is nonempty. Take $g=0$ on $E$ and $g=C_{2}f$ on $M\setminus E$ with the constant $C_{2}$ chosen so that $\int_{M}g\omega^{n}=\int_{M}\omega^{n}.$

Now we compare $u_{j,\delta}$ with

where the coordinates of $B^{\prime\prime}_{j}$ are used. Given $z\in B_{j}$ we find $w_{z}$ with $|w_{z}|=\delta$ such that

(Note that defining $\tilde{u}_{j,\sqrt{\delta}}(z)$ and $\tilde{u}_{j,\sqrt{\delta}}(z+w_{z})$ we integrate over the same domain except for the piece of volume at most $2\tau(n)\delta^{n+\frac{1}{2}}.$)

Since $\alpha<1/2$ we infer from this estimate for $\delta<\delta_{0}$ and $\delta_{0}$ small enough

Thus, as $||\Delta u||_{1}$ is a priori bounded on every $B^{\prime\prime}_{j}$, applying 1.1 one obtains

for all $j$ and consequently

with the constant depending only on $M$. Hence, upon the use of Hölder inequality

where $c_{5}$ depends also on $||f||_{p}$. By Theorem 1.1 for $\delta<\delta_{1}$ and $\delta_{1}$ small enough if $v$ solves

and is suitably normalized then

(since by our choice of $\alpha$ we have $\alpha<\frac{1-\alpha}{q(n+3+\epsilon)}$).

Applying the proposition one can find $c_{7}$ such that

By the comparison principle [K2] and 2.4

This contradiction shows that the choice of small enough $\delta$ with $\chi(\delta)>\max(9,\chi(N\delta)$ is impossible. Thus the proof of the claim and that of the theorem is completed.

References

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