A priori estimates for weak solutions of complex Monge-Amp\`ere equations

TL;DR

This paper develops a priori estimates for weak solutions of complex Monge-Ampère equations on compact Kähler manifolds, extending previous results and providing a unified framework that includes Yau's classical estimates.

Contribution

It introduces new a priori estimates for solutions in weighted energy classes, generalizes existing results, and offers a simplified proof of Yau's ${ m C}^0$-estimate.

Findings

Solutions are controlled by Monge-Ampère capacity estimates.

Measures dominated by capacity belong to the Monge-Ampère operator's range.

Unified approach extends classical results by Cegrell and Kolodziej.

Abstract

Let $X$ be a compact K\"ahler manifold and $\om$ a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal E}_{\chi}(X,\om)$ of $\om$-plurisubharmonic functions of finite weighted Monge-Amp\`ere energy. When the weight $\chi$ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Amp\`ere capacity, then it belongs to the range of the Monge-Amp\`ere operator on some class ${\mathcal E}_{\chi}(X,\om)$. This is done by establishing a priori estimates on the capacity of sublevel sets of the solutions. Our result extends U.Cegrell's and S.Kolodziej's results and puts them into a unifying frame. It also gives a simple proof of S.T.Yau's celebrated a priori ${\mathcal C}^0$-estimate.

Equations

$$\text{mbox}{(MA)}_{\mu }(\omega +d{d}^c\phi {)}^n=\mu ,$$

$$\mu (K)\le F(Ca{p}_{\omega }(K)),\text{mbox}forallBorelsubsetsK\subset X.$$

$$Ca{p}_{\omega }(\phi <-s)\le \exp (-n{H}^{-1}(s)).$$

$${\mathcal{C}}_A(\omega ):=\{\nu \text{probability measure }/\nu (K)\le A\cdot Ca{p}_{\omega }(K),\text{for all }K\subset X\},$$

$$Ca{p}_{\omega }(K):=\sup \left\{{\int }_K(\omega +d{d}^{c}u{)}^n/u\in PSH(X,\omega ),0\le u\le 1\right\}.$$

$$\mathcal{E}^{p}(X,\omega):=\{\psi\in\mathcal{E}(X,\omega)\ /\ \psi\in L^{p}\big{(}(\omega+dd^{c}\psi)^{n}\big{)}\}.$$

$$Ca{p}_{{\omega }_j}(v<-t)\le \frac{C_0}{t},\text{for all }t\ge 1.$$

$${\int }_X(-v{)}^{p}d\nu$$

$${\int }_X(-{\phi }_j{)}^p({\omega }_j+d{d}^c{\phi }_j{)}^n\le 2{\int }_X(-{\phi }_j{)}^{p}d\nu \le 2\left[\frac{pA{C}_0}{1-p}+C_p\right].$$

$${\int }_X(-{\varphi }_j{)}^p({\omega }_j+d{d}^c{\varphi }_j{)}^n\le 2^n{\int }_X(-{\phi }_j{)}^p({\omega }_j+d{d}^c{\phi }_j{)}^n\le C^{\prime }<+\infty .$$

$$({\omega }_j+d{d}^c{\varphi }_j{)}^n\ge \underset{l\ge j}{\inf }({\omega }_l+d{d}^c{\phi }_l{)}^n\ge \nu ,$$

$$\mu =f\cdot \nu ,\nu \in {\mathcal{C}}_1(\omega ),0\le f\in L^1(\nu ).$$

$${\mathcal{E}}_{\chi }(X,\omega ):=\left\{\phi \in \mathcal{E}(X,\omega )/{\int }_X(-\chi )(-|\phi |)(\omega +d{d}^c\phi {)}^n<+\infty \right\},$$

$$\underset{j}{\sup }{\int }_X(-\chi )\circ {\phi }_j(\omega +d{d}^c{\phi }_j{)}^n<+\infty ,\text{ where }{\phi }_j:=\max (\phi ,-j)$$

$${\hat{\mathcal{E}}}_{\chi }(X,\omega ):=\left\{\phi \in PSH(X,\omega )/{\int }_0^{+\infty }{t}^n{\chi }^{\prime }(-t)Ca{p}_{\omega }(\phi <-t)dt<+\infty \right\}.$$

$${\mathcal{E}}_{\hat{\chi }}(X,\omega )\subset {\hat{\mathcal{E}}}_{\chi }(X,\omega ),\text{ where }{\chi }^{\prime }(t-1)=t^n{\hat{\chi }}^{\prime }(t).$$

$$\left\{a\phi +(1-a)\psi <-t\right\}\subset \left\{\phi <-t\right\}\cup \left\{\psi <-t\right\}.$$

$${\int }_X(-\chi )\circ {\phi }_j(\omega +d{d}^c{\phi }_j{)}^n$$

$$PSH(X,\omega )\cap L^{\infty }(X)=\bigcap _{\chi }{\hat{\mathcal{E}}}_{\chi }(X,\omega ),$$

$$t^{n}Ca{p}_{\omega }(\phi <-s-t)\le {\int }_{(\phi <-s)}(\omega +d{d}^c\phi {)}^n\le s^{n}Ca{p}_{\omega }(\phi <-s),$$

$$\mu (X)=Vo{l}_{\omega }(X)={\int }_X{\omega }^n=1.$$

$$F_{\epsilon }(x):=x[\epsilon (-\ln x/n){]}^n,x>0.$$

$$\mu (K)\le F_{\epsilon }(Ca{p}_{\omega }(K)).$$

$$Ca{p}_{\omega }(\phi <-s)\le \exp (-n{H}^{-1}(s)),\text{ for all }s>0.$$

$$Ca{p}_{\omega }(\phi <-s)\lesssim \exp \left(-ns/e\right).$$

$$\mu =(\omega +d{d}^c\phi {)}^n\text{ with }\phi \in PSH(X,\omega )\cap L^{\infty }(X),$$

$$g(s):=-\frac{1}{n}\log Ca{p}_{\omega }(\phi <-s),\forall s>0.$$

$$g(0)=-\frac{1}{n}\log Ca{p}_{\omega }(X)=-\frac{1}{n}\log Vo{l}_{\omega }(X)=0.$$

$$t^{n}Ca{p}_{\omega }(\phi <-s-t)\le \mu (\phi <-s)\le F_{\epsilon }\left(Ca{p}_{\omega }(\phi <-s)\right).$$

Full text

We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class ${\mathcal{E}}_{\chi}(X,\omega)$. This is done by establishing a priori estimates on the capacity of sublevel sets of the solutions.

Our result extends U.Cegrell’s and S.Kolodziej’s results and puts them into a unifying frame. It also gives a simple proof of S.T.Yau’s celebrated a priori ${\mathcal{C}}^{0}$-estimate.

2000 Mathematics Subject Classification: 32W20, 32Q25, 32U05.

1. Introduction

Let $X$ be a compact connected Kähler manifold of dimension $n\in\mathbb{N}^{*}$. Throughout the article $\omega$ denotes a smooth closed form of bidegree $(1,1)$ which is nonnegative and big, i.e. such that $\int_{X}\omega^{n}>0$. We continue the study started in [GZ 2], [EGZ] of the complex Monge-Ampère equation

[TABLE]

where $\varphi$, the unknown function, is $\omega$-plurisubharmonic: this means that $\varphi\in L^{1}(X)$ is upper semi-continuous and $\omega+dd^{c}\varphi\geq 0$ is a positive current. We let $PSH(X,\omega)$ denote the set of all such functions (see [GZ 1] for their basic properties). Here $\mu$ is a fixed positive Radon measure of total mass $\mu(X)=\int_{X}\omega^{n}$, and $d=\partial+\overline{\partial}$, $d^{c}=\frac{1}{2i\pi}(\partial-\overline{\partial})$.

Following [GZ 2] we say that a $\omega$-plurisubharmonic function $\varphi$ has finite weighted Monge-Ampère energy, $\varphi\in{\mathcal{E}}(X,\omega)$, when its Monge-Ampère measure $(\omega+dd^{c}\varphi)^{n}$ is well defined, and there exists an increasing function $\chi:\ \mathbb{R}^{-}\to\mathbb{R}^{-}$ such that $\chi(-\infty)=-\infty$ and $\chi\circ\varphi\in L^{1}((\omega+dd^{c}\varphi)^{n})$. In general $\chi$ has very slow growth at infinity, so that $\varphi$ is far from being bounded.

The purpose of this article is twofold. First we extend one of the main results of [GZ 2] by showing

THEOREM A. *There exists $\varphi\in{\mathcal{E}}(X,\omega)$ such that $\mu=(\omega+dd^{c}\varphi)^{n}$ if and only if $\mu$ does not charge pluripolar sets. *

This results has been established in [GZ 2] when $\omega$ is a Kähler form. It is important for applications to complex dynamics and Kähler geometry to consider as well forms $\omega$ that are less positive (see [EGZ]).

We then look for conditions on the measure $\mu$ which insure that the solution $\varphi$ is almost bounded. Following the seminal work of S. Kolodziej [K 2,3], we say that $\mu$ is dominated by the Monge-Ampère Capacity $Cap_{\omega}$ if there exists a function $F:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that $\lim_{t\to 0^{+}}F(t)=0$ and

[TABLE]

Here $Cap_{\omega}$ denotes the global version of the Monge-Ampère capacity introduced by E.Bedford and A.Taylor [BT] (see section 2).

Observe that $\mu$ does not charge pluripolar sets since $F(0)=0.$ When $F(x)\lesssim x^{\alpha}$ vanishes at order $\alpha>1$ and $\omega$ is Kähler, S. Kolodziej has proved [K 2] that the solution $\varphi\in PSH(X,\omega)$ of (MA)μ is continuous. The boundedness part of this result was extended in [EGZ] to the case when $\omega$ is merely big and nonnegative. If $F(x)\lesssim x^{\alpha}$ with $0<\alpha<1,$ two of us have proved in [GZ 2] that the solution $\varphi$ has finite $\chi-$energy, where $\chi(t)=-(-t)^{p},\ p=p(\alpha)>0$. This result was first established by U. Cegrell in a local context [Ce].

Another objective of this article is to fill in the gap inbetween Cegrell’s and Kolodziej’s results, by considering all intermediate dominating functions $F.$ Write $F_{\varepsilon}(x)=x[\varepsilon(-\ln(x)/n)]^{n}$ where $\varepsilon:\mathbb{R}\rightarrow[0,\infty[$ is nonincreasing. Our second main result is:

THEOREM B. If $\mu(K)\leq F_{\varepsilon}(Cap_{\omega}(K))$ for all Borel subsets $K\subset X$, then $\mu=(\omega+dd^{c}\varphi)^{n}$ where $\varphi\in PSH(X,\omega)$ satisfies $\sup_{X}\varphi=0$ and

[TABLE]

*Here $H^{-1}$ is the reciprocal function of $H(x)=e\int_{0}^{x}\varepsilon(t)dt+s_{0},$ where $s_{0}=s_{0}(\varepsilon,\omega)\geq 0$ only depends on $\varepsilon$ and $\omega.$ *

This general statement has several useful consequences:

• •

if $\int_{0}^{+\infty}\varepsilon(t)dt<+\infty,$ then $H^{-1}(s)=+\infty$ for $s\geq s_{\infty}:=e\int_{0}^{+\infty}\varepsilon(t)dt+s_{0},$ hence $Cap_{\omega}(\varphi<-s)=0.$ This means that $\varphi$ is bounded from below by $-s_{\infty}.$ This result is due to S. Kolodziej [K 2,3] when $\omega$ is Kähler, and [EGZ] when $\omega\geq 0$ is merely big;

• •

the condition (†) is easy to check for measures with density in $L^{p}$, $p>1$. Our result thus gives a simple proof (Corollary 3.2), following the seminal approach of S. Kolodziej ([K2]), of the $\mathcal{C}^{0}$-a priori estimate of S.T. Yau [Y], which is crucial for proving the Calabi conjecture (see [T] for an overview);

• •

when $\int_{0}^{+\infty}\varepsilon(t)dt=+\infty,$ the solution $\varphi$ is generally unbounded. The faster $\varepsilon(t)$ decreases towards zero, the faster the growth of $H^{-1}$ at infinity, hence the closer is $\varphi$ from being bounded;

• •

the special case $\varepsilon\equiv 1$ is of particular interest. Here $\mu(\cdot)\leq Cap_{\omega}(\cdot)$, and our result shows that $Cap_{\omega}(\varphi<-s)$ decreases exponentially fast, hence $\varphi$ has “ loglog-singularities”. These are the type of singularities of the metrics used in Arakelov geometry in relation with measures $\mu=fdV$ whose density has Poincaré-type singularities (see [Ku], [BKK]).

We prove Theorem B in section 3, after establishing Theorem A in section 2.1 and recalling some useful facts from [GZ 2], [EGZ] in section 2.2. We then test the sharpness of our estimates in section 4, where we give examples of measures fulfilling our assumptions: these are absolutely continuous with respect to $\omega^{n}$, and their density do not belong to $L^{p}$, for any $p>1$.

2. Weakly singular quasiplurisubharmonic functions

The class ${\mathcal{E}}(X,\omega)$ of $\omega$-psh functions with finite weighted Monge-Ampère energy has been introduced and studied in [GZ 2]. It is the largest subclass of $PSH(X,\omega)$ on which the complex Monge-Ampère operator $(\omega+dd^{c}\cdot)^{n}$ is well-defined and the comparison principle is valid. Recall that $\varphi\in{\mathcal{E}}(X,\omega)$ if and only if $(\omega+dd^{c}\varphi_{j})^{n}(\varphi\leq-j)\rightarrow 0$, where $\varphi_{j}:=\max(\varphi,-j)$.

2.1. The range of the Monge-Ampère operator

The range of the operator $(\omega+dd^{c}\cdot)^{n}$ acting on ${\mathcal{E}}(X,\omega)$ has been characterized in [GZ 2] when $\omega$ is a Kähler form. We extend here this result to the case when $\omega$ is merely nonnegative and big.

Theorem 2.1**.**

Assume $\omega$ is a smooth closed nonnegative (1,1) form on $X$, and $\mu$ is a positive Radon measure such that $\mu(X)=\int_{X}\omega^{n}>0$.

Then there exists $\varphi\in{\mathcal{E}}(X,\omega)$ such that $\mu=(\omega+dd^{c}\varphi)^{n}$ if and only if $\mu$ does not charge pluripolar sets.

Proof.

We can assume without loss of generality that $\mu$ and $\omega$ are normalized so that $\mu(X)=\int_{X}\omega^{n}=1.$ Consider, for $A>0$,

[TABLE]

where $Cap_{\omega}$ denotes the Monge-Ampère capacity introduced by E.Bedford and A.Taylor in [BT] (see [GZ 1] for this compact setting). Recall that

[TABLE]

We first show that a measure $\nu\in\mathcal{C}_{A}(\omega)$ is the Monge-Ampère of a function $\psi\in\mathcal{E}^{p}(X,\omega),$ for any $0<p<1$, where

[TABLE]

Indeed, fix $\nu\in\mathcal{C}_{A}(\omega),\ 0<p<1,$ and $\omega_{j}:=\omega+\varepsilon_{j}\Omega$, where $\Omega$ is a kähler form on $X$, and $\varepsilon_{j}>0$ decreases towards zero. Observe that $PSH(X,\omega)\subset PSH(X,\omega_{j}),$ hence $Cap_{\omega}(.)\leq Cap_{\omega_{j}}(.),$ so that $\nu\in\mathcal{C}_{A}(\omega_{j}).$ It follows from Proposition 3.6 and 2.7 in [GZ 1] that there exists $C_{0}>0$ such that for any $v\in PSH(X,\omega_{j})$ normalized by $\sup_{X}v=-1,$ we have

[TABLE]

This yields $\mathcal{E}^{p}(X,\omega_{j})\subset L^{p}(\nu)$: if $v\in\mathcal{E}^{p}(X,\omega_{j})$ with $\sup_{X}v=-1,$ then

[TABLE]

It follows therefore from Theorem 4.2 in [GZ 2] that there exists $\varphi_{j}\in\mathcal{E}^{p}(X,\omega_{j})$ with $\sup_{X}\varphi_{j}=-1$ and $(\omega_{j}+dd^{c}\varphi_{j})^{n}=c_{j}\cdot\nu,$ where $c_{j}=\int_{X}\omega_{j}^{n}\geq 1$ decreases towards 1 as $\varepsilon_{j}$ decreases towards zero. We can assume without loss of generality that $1\leq c_{j}\leq 2.$ Observe that the $\varphi_{j}$’s have uniformly bounded energies, namely

[TABLE]

Since $\sup_{X}\varphi_{j}=-1,$ we can assume (after extracting a convergent subsequence) that $\varphi_{j}\to\varphi$ in $L^{1}(X),$ where $\varphi\in PSH(X,\omega),\ \sup_{X}\varphi=-1.$

Set $\phi_{j}:=(\sup_{l\geq j}\varphi_{l})^{*}.$ Thus $\phi_{j}\in PSH(X,\omega_{j}),$ and $\phi_{j}$ decreases towards $\varphi.$ Since $\phi_{j}\geq\varphi_{j},$ it follows from the “fundamental inequality” (Lemma 2.3 in [GZ 2]) that

[TABLE]

Hence it follows from stability properties of the class $\mathcal{E}^{p}(X,\omega)$ that $\varphi\in\mathcal{E}^{p}(X,\omega)$ (see Proposition 5.6 in [GZ 2]). Moreover

[TABLE]

hence $(\omega+dd^{c}\varphi)^{n}=\lim(\omega_{j}+dd^{c}\phi_{j})^{n}\geq\nu.$ Since $\int_{X}\omega^{n}=\nu(X)=1,$ this yields $\nu=(\omega+dd^{c}\varphi)^{n}$ as claimed above.

We can now prove the statement of the theorem. One implication is obvious: if $\mu=(\omega+dd^{c}\varphi)^{n},\ \varphi\in\mathcal{E}(X,\omega),$ then $\mu$ does not charge pluripolar sets, as follows from Theorem 1.3 in [GZ 2].

So we assume now $\mu$ that does not charge pluripolar sets. Since $\mathcal{C}_{1}(\omega)$ is a compact convex set of probability measures which contains all measures $(\omega+dd^{c}u)^{n}$, $u\in PSH(X,\omega),\ 0\leq u\leq 1,$ we can project $\mu$ onto $\mathcal{C}_{1}(\omega)$ and get, by a generalization of Radon-Nikodym theorem (see [R], [Ce]),

[TABLE]

Now $\nu=(\omega+dd^{c}\psi)^{n}$ for some $\psi\in\mathcal{E}^{1/2}(X,\omega),\ \psi\leq 0,$ as follows from the discussion above. Replacing $\psi$ by $e^{\psi}$ shows that we can actually assume $\psi$ to be bounded (see Lemma 4.5 in [GZ 2]). We can now apply line by line the same proof as that of Theorem 4.6 in [GZ 2] to conclude that $\mu=(\omega+dd^{c}\varphi)^{n}$ for some $\varphi\in\mathcal{E}(X,\omega).$ ∎

2.2. High energy and capacity estimates

Given $\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ an increasing function, we consider, following [GZ 2],

[TABLE]

Alternatively a function $\varphi\leq 0$ belongs to ${\mathcal{E}_{\chi}(X,\omega)}$ if and only if

[TABLE]

is the canonical approximation of $\varphi$ by bounded $\omega$-psh functions. When $\chi(t)=-(-t)^{p}$, ${\mathcal{E}_{\chi}(X,\omega)}$ is the class ${\mathcal{E}}^{p}(X,\omega)$ used in previous section.

The properties of classes ${\mathcal{E}_{\chi}(X,\omega)}$ are quite different whether the weight $\chi$ is convex (slow growth at infinity) or concave. In previous works [GZ 2], two of us were mainly interested in weights $\chi$ of moderate growth at infinity (at most polynomial). Our main objective in the sequel is to construct solutions $\varphi$ of $(MA)_{\mu}$ which are “almost bounded”, i.e. in classes ${\mathcal{E}_{\chi}(X,\omega)}$ for concave weights $\chi$ of arbitrarily high growth.

For this purpose it is useful to relate the property $\varphi\in{\mathcal{E}_{\chi}(X,\omega)}$ to the speed of decreasing of $Cap_{\omega}(\varphi<-t)$, as $t\rightarrow+\infty$. We set

[TABLE]

An important tool in the study of classes ${\mathcal{E}_{\chi}(X,\omega)}$ are the “fundamental inequalities” (Lemmas 2.3 and 3.5 in [GZ 2]), which allow to compare the weighted energy of two $\omega$-psh functions $\varphi\leq\psi$. These inequalities are only valid for weights of slow growth (at most polynomial), while they become immediate for classes $\hat{{\mathcal{E}}}_{\chi}(X,\omega)$. So are the convexity properties of $\hat{{\mathcal{E}}}_{\chi}(X,\omega)$. We summarize this and compare these classes in the following:

Proposition 2.2**.**

The classes $\hat{{\mathcal{E}}}_{\chi}(X,\omega)$ are convex and stable under maximum: if $\hat{{\mathcal{E}}}_{\chi}(X,\omega)\ni\varphi\leq\psi\in PSH(X,\omega)$, then $\psi\in\hat{{\mathcal{E}}}_{\chi}(X,\omega)$.

One always has $\hat{{\mathcal{E}}}_{\chi}(X,\omega)\subset{\mathcal{E}_{\chi}(X,\omega)}$, while

[TABLE]

Since we are mainly interested in the sequel in weights with (super) fast growth at infinity, the previous proposition shows that $\hat{{\mathcal{E}}}_{\chi}(X,\omega)$ and ${\mathcal{E}_{\chi}(X,\omega)}$ are roughly the same: a function $\varphi\in PSH(X,\omega)$ belongs to one of these classes if and only if $Cap_{\omega}(\varphi<-t)$ decreases fast enough, as $t\rightarrow+\infty$.

Proof.

The convexity of $\hat{{\mathcal{E}}}_{\chi}(X,\omega)$ follows from the following simple observation: if $\varphi,\psi\in\hat{{\mathcal{E}}}_{\chi}(X,\omega)$ and $0\leq a\leq 1$, then

[TABLE]

The stability under maximum is obvious.

Assume $\varphi\in\hat{{\mathcal{E}}}_{\chi}(X,\omega)$. We can assume without loss of generality $\varphi\leq 0$ and $\chi(0)=0$. Set $\varphi_{j}:=\max(\varphi,-j)$. It follows from Lemma 2.3 below that

[TABLE]

This shows that $\varphi\in{\mathcal{E}_{\chi}(X,\omega)}$. The other inclusion goes similarly, using the second inequality in Lemma 2.3 below. ∎

If $\varphi\in{\mathcal{E}_{\chi}(X,\omega)}$ (or $\hat{{\mathcal{E}}}_{\chi}(X,\omega)$), then the bigger the growth of $\chi$ at $-\infty$, the smaller $Cap_{\omega}(\varphi<-t)$ when $t\rightarrow+\infty$, hence the closer $\varphi$ is from being bounded. Indeed $\varphi\in PSH(X,\omega)$ is bounded iff it belongs to ${\mathcal{E}_{\chi}(X,\omega)}$ for all weights $\chi$, as was observed in [GZ 2], Proposition 3.1. Similarly

[TABLE]

where the intersection runs over all concave increasing functions $\chi$.

We will make constant use of the following result:

Lemma 2.3**.**

Fix $\varphi\in{\mathcal{E}}(X,\omega)$. Then for all $s>0$ and $0\leq t\leq 1$,

[TABLE]

where the second inequality is true only for $s\geq 1$.

The proof is a direct consequence of the comparison principle (see Lemma 2.2 in [EGZ] and [GZ 2]).

3. Measures dominated by capacity

From now on $\mu$ denotes a positive Radon measure on $X$ whose total mass is $Vol_{\omega}(X)$: this is an obvious necessary condition in order to solve $(MA)_{\mu}$. To simplify numerical computations, we assume in the sequel that $\mu$ and $\omega$ have been normalized so that

[TABLE]

When $\mu=e^{h}\omega^{n}$ is a smooth volume form and $\omega$ is a Kähler form, S.T.Yau has proved [Y] that $(MA)_{\mu}$ admits a unique smooth solution $\varphi\in PSH(X,\omega)$ with $\sup_{X}\varphi=0$. Smooth measures are easily seen to be nicely dominated by the Monge-Ampère capacity (see the proof of Corollary 3.2 below).

Measures dominated by the Monge-Ampère capacity have been extensively studied by S.Kolodziej in [K 2,3,4]. Following S. Kolodziej ([K3], [K4]) with slightly different notations, fix $\varepsilon:\mathbb{R}\rightarrow[0,\infty[$ a continuous decreasing function and set

[TABLE]

We will consider probability measures $\mu$ satisfying the following condition : for all Borel subsets $K\subset X$,

[TABLE]

The main result achieved in [K 2], can be formulated as follows: If $\omega$ is a Kähler form and $\int_{0}^{+\infty}{\varepsilon(t)}dt<+\infty$ then $\mu=(\omega+dd^{c}\varphi)^{n}$ for some continuous function $\varphi\in PSH(X,\omega)$.

The condition $\int_{0}^{+\infty}{\varepsilon(t)}dt<+\infty$ means that $\varepsilon$ decreases fast enough towards zero at infinity. This gives a quantitative estimate on how fast $\varepsilon(-\ln Cap_{\omega}(K)/n)$, hence $\mu(K)$, decreases towards zero as $Cap_{\omega}(K)\rightarrow 0$.

When $\int_{0}^{+\infty}\varepsilon(t)dt=+\infty$, it follows from Theorem 2.1 that $\mu=(\omega+dd^{c}\varphi)^{n}$ for some function $\varphi\in{\mathcal{E}}(X,\omega)$, but $\varphi$ will generally be unbounded. Our second main result measures how far $\varphi$ is from being bounded:

Theorem 3.1**.**

Assume for all compact subsets $K\subset X$,

[TABLE]

Then $\mu=(\omega+dd^{c}\varphi)^{n}$ where $\varphi\in{\mathcal{E}}(X,\omega)$ is such that $\sup_{X}\varphi=0$ and

[TABLE]

Here $H^{-1}$ is the reciprocal function of $H(x)=e\int_{0}^{x}\varepsilon(t)dt+s_{0}$, where $s_{0}=s_{0}(\varepsilon,\omega)\geq 0$ is a constant which only depends on $\varepsilon$ and $\omega.$

In particular $\varphi\in{\mathcal{E}_{\chi}(X,\omega)}$ where $-\chi(-t)=\exp(nH^{-1}(t)/2)$.

Recall that here, and troughout the article, $\omega\geq 0$ is merely big.

Before proving this result we make a few observations.

• •

It is interesting to consider as well the case when $\varepsilon(t)$ increases towards $+\infty$. One can then obtain solutions $\varphi$ such that $Cap_{\omega}(\varphi<-t)$ decreases at a polynomial rate. When e.g. $\omega$ is Kähler and $\mu(K)\leq Cap_{\omega}(K)^{\alpha}$, $0<\alpha<1$, it follows from Proposition 5.3 in [GZ 2] that $\mu=(\omega+dd^{c}\varphi)^{n}$ where $\varphi\in{\mathcal{E}}^{p}(X,\omega)$ for some $p=p_{\alpha}>0$. Here ${\mathcal{E}}^{p}(X,\omega)$ denotes the Cegrell type class ${\mathcal{E}}_{\chi}(X,\omega),$ with $\chi(t)=-(-t)^{p}.$

• •

When $\varepsilon(t)\equiv 1$, $F_{\varepsilon}(x)=x$ and $H(x)\asymp e.x$. Thus Theorem 3.1 reads $\mu\leq Cap_{\omega}\Rightarrow\mu=(\omega+dd^{c}\varphi)^{n}$, where

[TABLE]

This is precisely the rate of decreasing corresponding to functions which look locally like $-\log(-\log||z||)$, in some local chart $z\in U\subset\mathbb{C}^{n}$. This class of $\omega$-psh functions with “loglog-singularities” is important for applications (see [Ku], [BKK]).

• •

If $\varepsilon(t)$ decreases towards zero, then $Cap_{\omega}(\varphi<-t)$ decreases at a superexponential rate. The faster $\varepsilon(t)$ decreases towards zero, the slower the growth of $H$, hence the faster the growth of $H^{-1}$ at infinity. When $\int^{+\infty}\varepsilon(t)dt<+\infty$, the function $\varepsilon$ decreases so fast that $Cap_{\omega}(\varphi<-t)=0$ for $t>>1$, thus $\varphi$ is bounded. This is the case when $\mu(K)\leq Cap_{\omega}(K)^{\alpha}$ for some $\alpha>1$ [K 2], [EGZ].

• •

When $\int^{+\infty}\varepsilon(t)dt=+\infty$, the solution $\varphi$ may well be unbounded (see Examples in section 4). At the critical case where $\mu\leq F_{\varepsilon}(Cap_{\omega})$ for all functions $\varepsilon$ such that $\int^{+\infty}\varepsilon(t)dt=+\infty$, we obtain

[TABLE]

as follows from Proposition 3.1 in [GZ 2]. This partially explains the difficulty in describing the range of Monge-Ampère operators on the set of bounded (quasi-)psh functions.

Proof.

The assumption on $\mu$ implies in particular that it vanishes on pluripolar sets. It follows from Theorem 2.1 that there exists a function $\varphi\in\mathcal{E}(X,\omega)$ such that $\mu=(\omega+dd^{c}\varphi)^{n}$ and $\sup_{X}\varphi=0.$ Set

[TABLE]

The function $g$ is increasing on $[0,+\infty]$ and $g(+\infty)=+\infty$, since $Cap_{\omega}$ vanishes on pluripolar sets. Observe also that $g(s)\geq 0$ for all $s\geq 0$, since

[TABLE]

It follows from Lemma 2.3 and (3.1) that for all $s>0$ and $0\leq t\leq 1$,

[TABLE]

Therefore for all $s>0$ and $0\leq t\leq 1$,

[TABLE]

We define an increasing sequence $(s_{j})_{j\in\mathbb{N}}$ by induction setting

[TABLE]

The choice of $s_{0}$. Recall that (3.2) is only valid for $0\leq t\leq 1.$ We choose $s_{0}\geq 0$ large enough so that

[TABLE]

This will allow us to use (3.2) with $t=t_{j}=s_{j+1}-s_{j}\in[0,1]$, since $\varepsilon\circ g$ is decreasing, while $s_{j}\geq s_{0}$ is increasing, hence

[TABLE]

We must insure that $s_{0}=s_{0}(\varepsilon,\omega)$ can chosen to be independent of $\varphi.$ This is a consequence of Proposition 2.7 in [GZ 1]: since $\sup_{X}\varphi=0,$ there exists $c_{1}(\omega)>0$ so that $0\leq\int_{X}(-\varphi)\omega^{n}\leq c_{1}(\omega),$ hence

[TABLE]

Therefore $g(s_{0})\geq\varepsilon^{-1}(1/e)$ for $s_{0}=s_{0}(\varepsilon,\omega):=(n+c_{1}(\omega))\exp(n\varepsilon^{-1}(1/e)),$ which is independent of $\varphi$. This yields $e.\varepsilon\circ g(s_{0})\leq 1$, as desired.

*The growth of $s_{j}$. * We can now apply (3.2) and get $g(s_{j})\geq j+g(s_{0})\geq j.$ Thus $\lim g(s_{j})=+\infty$. There are two cases to be considered.

If $s_{\infty}=\lim s_{j}\in\mathbb{R}^{+}$, then $g(s)\equiv+\infty$ for $s>s_{\infty}$, i.e. $Cap_{\omega}(\varphi<-s)=0,\ \ \forall s>s_{\infty}$. Therefore $\varphi$ is bounded from below by $-s_{\infty}$, in particular $\varphi\in\mathcal{E}_{\chi}(X,\omega)$ for all $\chi.$

Assume now (second case) that $s_{j}\to+\infty.$ For each $s>0,$ there exists $N=N_{s}\in\mathbb{N}$ such that $s_{N}\leq s<s_{N+1}.$ We can estimate $s\mapsto N_{s}$:

[TABLE]

Therefore $H^{-1}(s)\leq N\leq g(s_{N})\leq g(s),$ hence

[TABLE]

Set now $-\chi(-t)=\exp(nH^{-1}(t)/2)$. Then

[TABLE]

This shows that $\varphi\in{\mathcal{E}_{\chi}(X,\omega)}$ where $\chi(t)=-\exp(nH^{-1}(-t)/2)$.

It follows from the proof above that when $\int_{0}^{+\infty}\varepsilon(t)dt<+\infty$, the solution $\varphi$ is bounded since in this case we have

[TABLE]

where $s_{0}(\varepsilon,\omega)$ is an absolute constant satisfying $(3.3)$ (see above). ∎

Let us emphasize that Theorem 3.1 also yields a slightly simplified proof of the following result [K 2], [EGZ]: if $\mu(K)\leq F_{\varepsilon}(Cap_{\omega}(K))$ for some decreasing function $\varepsilon:\mathbb{R}\rightarrow\mathbb{R}^{+}$ such that $\int^{+\infty}\varepsilon(t)dt<+\infty$, then the sequence $(s_{j})$ above is convergent, hence $\mu=(\omega+dd^{c}\varphi)^{n}$, where $\varphi\in PSH(X,\omega)$ is bounded. For the reader’s convenience we indicate a proof of the following important particular case:

Corollary 3.2**.**

Let $\mu=f\omega^{n}$ be a measure with density $0\leq f\in L^{p}(\omega^{n})$, where $p>1$ and $\int_{X}f\omega^{n}=\int_{X}\omega^{n}$. Then there exists a unique bounded function $\varphi\in PSH(X,\omega)$ such that $(\omega+dd^{c}\varphi)^{n}=\mu$, $\sup_{X}\varphi=0$ and

[TABLE]

where $C(p,\omega)>0$ only depends on $p$ and $\omega$.

This a priori bound is a crucial step in the proof by S.T.Yau of the Calabi conjecture (see [Ca], [Y], [A], [T], [Bl]). The proof presented here follows Kolodziej’s new and decisive pluripotential approach (see [K2]). Let us stress that the dependence $\omega\longmapsto C(p,\omega)$ is quite explicit, as we shall see in the proof. This is important when considering degenerate situations [EGZ].

Proof.

We claim that there exists $C_{1}(\omega)$ such that

[TABLE]

Assuming this for the moment, we can apply Theorem 3.1 with $\varepsilon(x)=C_{1}(\omega)||f||^{1/n}_{L^{p}(\omega^{n})}\exp(-x)$, which yields, as observed at the end of the proof of Theorem 3.1

[TABLE]

where $M(f,\omega):=s_{0}(\varepsilon,\omega)+e\ \varepsilon(0)+e\int_{0}^{+\infty}\varepsilon(t)dt=s_{0}(\varepsilon,\omega)+2eC_{1}(\omega)||f||^{1/n}_{L^{p}(\omega^{n})}$ and $s_{0}=s_{0}(\varepsilon,\omega)$ is a large number $s_{0}>1$ satisfying the inequality $(3.3)$.

In order to give the precise dependence of the uniform bound $M(f,\omega)$ on the $L^{p}-$norm of the density $f$, we need to choose $s_{0}$ more carefully. Observe that condition $(3.3)$ can be written

[TABLE]

Since n\varepsilon^{-1}(1/e)=\log\Bigl{(}e^{n}C_{1}(\omega)^{n}\|f\|_{L^{p}(\omega^{n})}\Bigr{)}, we must choose $s_{0}>0$ so that

[TABLE]

We claim that for any $N\geq 1$ there exists a uniform constant $C_{2}(N,p,\omega)>0$ such that for any $s>0,$

[TABLE]

Indeed observe first that by Hölder inequality,

[TABLE]

Since $\varphi$ belongs to the compact family $\{\psi\in PSH(X,\omega);\sup_{X}\psi=0\}$ ([GZ2]), there exists a uniform constant $C^{\prime}_{2}(N,p,\omega)>0$ such that $\|\varphi\|_{L^{Nq}(\omega^{n})}^{N}\leq C^{\prime}_{2}(N,p,\omega)$, hence

[TABLE]

Fix $u\in PSH(X,\omega)$ with $-1\leq u\leq 0$ and $N\geq 1$ to be specified later. If follows from Tchebysheff and energy inequalities ([GZ2]) that

[TABLE]

We have used here the fact that $\|f\|_{L^{p}(\omega^{n})}\geq 1$, which follows from the normalization : $1=\int_{X}\omega^{n}=\int_{X}f\omega^{n}\leq\|f\|_{L^{p}(\omega^{n})}$. This proves the claim.

Set $N=2n$, it follows from $(3.6)$ that $s_{0}:=C_{1}(\omega)^{n}e^{n}C_{2}(2n,p,\omega)\|f\|_{L^{p}(\omega^{n})}^{1/n}$ satisfies the required condition $(3.5)$, which implies the estimate of the theorem. We now establish the estimate $(3.4)$. Observe first that Hölder’s inequality yields

[TABLE]

Thus it suffices to estimate the volume $Vol_{\omega}(K)$. Recall the definition of the Alexander-Taylor capacity, $T_{\omega}(K):=\exp(-\sup_{X}V_{K,\omega})$, where

[TABLE]

This capacity is comparable to the Monge-Ampère capacity, as was observed by H.Alexander and A.Taylor [AT] (see Proposition 7.1 in [GZ 1] for this compact setting):

[TABLE]

It thus remains to show that $Vol_{\omega}(K)$ is suitably bounded from above by $T_{\omega}(K)$. This follows from Skoda’s uniform integrability result: set

[TABLE]

where $\nu(\psi,x)$ denotes the Lelong number of $\psi$ at point $x$. This actually only depends on the cohomology class $\{\omega\}\in H^{1,1}(X,\mathbb{R})$. It is a standard fact that goes back to H.Skoda (see [Z]) that there exists $C_{2}(\omega)>0$ so that

[TABLE]

for all functions $\psi\in PSH(X,\omega)$ normalized by $\sup_{X}\psi=0$. We infer

[TABLE]

It now follows from (3.7), (3.8), (3.9), that

[TABLE]

The conclusion follows by observing that $\exp(-1/x^{1/n})\leq C_{n}x^{2}$ for some explicit constant $C_{n}>0$. ∎

4. Examples

4.1. Measures invariant by rotations

In this section we produce examples of radially invariant functions/measures which show that our previous results are essentially sharp. The first example is due to S.Kolodziej [K 1].

Example 4.1**.**

We work here on the Riemann sphere $X=\mathbb{P}^{1}(\mathbb{C})$, with $\omega=\omega_{FS}$, the Fubini-Study volume form. Consider $\mu=f\omega$ a measure with density $f$ which is smooth and positive on $X\setminus\{p\}$, and such that

[TABLE]

in a local chart near $p=0$. A simple computation yields $\mu=\omega+dd^{c}\varphi$, where $\varphi\in PSH(\mathbb{P}^{1},\omega)$ is smooth in $\mathbb{P}^{1}\setminus\{p\}$ and $\varphi(z)\simeq-c^{\prime}\log(-\log|z|)$ near $p=0$, $c^{\prime}>0$, hence

[TABLE]

Here $a\simeq b$ means that $a/b$ is bounded away from zero and infinity.

This is to be compared to our estimate $\log Cap_{\omega}(\varphi<-t)\lesssim-t/e$ (Theorem 3.1 ) which can be applied, as it was shown by S.Kolodziej in [K 1] that $\mu\lesssim Cap_{\omega}$. Thus Theorem 3.1 is essentially sharp when $\varepsilon\equiv 1$.

We now generalize this example and show that the estimate provided by Theorem 3.1 is essentially sharp in all cases.

Example 4.2**.**

Fix $\varepsilon$ as in Theorem 3.1. Consider $\mu=f\omega$ on $X=\mathbb{P}^{1}(\mathbb{C})$, where $\omega=\omega_{FS}$ is the Fubini-Study volume form, $f\geq 0$ is continuous on $\mathbb{P}^{1}\setminus\{p\}$, and

[TABLE]

in local coordinates near $p=0$. Here $\varepsilon:\mathbb{R}\rightarrow\mathbb{R}^{+}$ decreases towards [math] at $+\infty$. We claim that there exists $A>0$ such that

[TABLE]

This is clear outside a small neighborhood of $p=0$ since the measure $\mu$ is there dominated by a smooth volume form. So it suffices to establish this estimate when $K$ is included in a local chart near $p=0$. Consider

[TABLE]

It is a classical fact (see e.g. [Ra]) that the logarithmic capacity $c(K)$ of $K$ can be estimated from below by the length of $\tilde{K}$, namely

[TABLE]

Using that $\varepsilon$ is decreasing, hence $0\leq-\varepsilon^{\prime}$, we infer

[TABLE]

Recall now that the logarithmic capacity c(K) is equivalent to Alexander-Taylor’s capacity $T_{\Delta}(K)$, which in turn is equivalent to the global Alexander-Taylor capacity $T_{\omega}(K)$ (see [GZ 1]): $c(K)\simeq T_{\Delta}(K)\simeq T_{\omega}(K).$ The Alexander-Taylor’s comparison theorem [AT] reads

[TABLE]

thus $\mu(K)\leq ACap_{\omega}(K)\varepsilon(-\log Cap_{\omega}(K))$.

We can therefore apply Theorem 3.1. It guarantees that $\mu=(\omega+dd^{c}\varphi)$, where $\varphi\in PSH(\mathbb{P}^{1},\omega)$ satisfies $\log Cap_{\omega}(\varphi<-s)\simeq-nH^{-1}(s)$, with $H(s)=eA\int_{0}^{s}{\varepsilon(t)}dt+s_{0}$. On the other hand a simple computation shows that $\varphi$ is continuous in $\mathbb{P}^{1}\setminus\{p\}$ and

[TABLE]

The sublevel set $(\varphi<-t)$ therefore coincides with the ball of radius $\exp(-\exp(H^{-1}(t)))$, hence $\log Cap_{\omega}(\varphi<-s)\simeq-H^{-1}(s).$

4.2. Measures with density

Here we consider the case when $\mu=fdV$ is absolutely continuous with respect to a volume form.

Proposition 4.3**.**

Assume $\mu=f\omega^{n}$ is a probability measure whose density satisfies $f[\log(1+f)]^{n}\in L^{1}(\omega^{n})$. Then $\mu\lesssim Cap_{\omega}$.

More generally if $f[\log(1+f)/\varepsilon(\log(1+|\log f|))]^{n}\in L^{1}(\omega^{n})$ for some continuous decreasing function $\varepsilon:\mathbb{R}\rightarrow\mathbb{R}^{+}_{*}$, then for all $K\subset X$,

[TABLE]

Proof.

With slightly different notations, the proof is identical to that of Lemma 4.2 in [K 4] to which we refer the reader. ∎

We now give examples showing that Proposition 4.3 is almost optimal.

Example 4.4**.**

For simplicity we give local examples. The computations to follow can also be performed in a global compact setting.

Consider $\varphi(z)=-\log(-\log||z||)$, where $||z||=\sqrt{|z_{1}|^{2}+\ldots+|z_{n}|^{2}}$ denotes the Euclidean norm in $\mathbb{C}^{n}$. One can check that $\varphi$ is plurisubharmonic in a neighborhood of the origin in $\mathbb{C}^{n}$, and that there exists $c_{n}>0$ so that

[TABLE]

Observe that $f[\log(1+f)]^{n-\alpha}\in L^{1},\ \forall\alpha>0$ but $f[\log(1+f)]^{n}\not\in L^{1}.$

When $n=1$ it was observed by S. Kolodziej [K 1] that $\mu(K)\lesssim Cap_{\omega}(K).$ Proposition 4.3 yields here

[TABLE]

For $n\geq 1,$ it follows from Proposition 4.3 and Theorem 3.1 that

[TABLE]

On the other hand, one can directly check that $\log Cap_{\omega}(\varphi<-s)\simeq-nH^{-1}(s).$

One can get further examples by considering $\varphi(z)=\chi\circ\log||z||$, so that

[TABLE]