A priori estimates for weak solutions of complex Monge-Amp\`ere
  equations
A PRIORI ESTIMATES FOR WEAK SOLUTIONS OF
COMPLEX MONGE-AMPÈRE EQUATIONS
S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI
Abstract. Let X be a compact Kähler manifold and ω a smooth
closed form of bidegree (1, 1) which is nonnegative and big. We study
the classes Eχ(X,ω) of ω-plurisubharmonic functions of finite weighted
Monge-Ampère energy. When the weight χ 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-Ampère capacity, then it belongs to the range of the Monge-
Ampère operator on some class Eχ(X,ω). 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 C0-estimate.
2000 Mathematics Subject Classification: 32W20, 32Q25, 32U05.
1. Introduction
Let X be a compact connected Kähler manifold of dimension n ∈ N∗.
Throughout the article ω denotes a smooth closed form of bidegree (1, 1)
which is nonnegative and big, i.e. such that
ωn > 0. We continue the
study started in [GZ 2], [EGZ] of the complex Monge-Ampère equation
(MA)µ (ω + dd
cϕ)n = µ,
where ϕ, the unknown function, is ω-plurisubharmonic: this means that
ϕ ∈ L1(X) is upper semi-continuous and ω+ ddcϕ ≥ 0 is a positive current.
We let PSH(X,ω) denote the set of all such functions (see [GZ 1] for their
basic properties). Here µ is a fixed positive Radon measure of total mass
µ(X) =
ωn, and d = ∂ + ∂, dc = 1
(∂ − ∂).
Following [GZ 2] we say that a ω-plurisubharmonic function ϕ has fi-
nite weighted Monge-Ampère energy, ϕ ∈ E(X,ω), when its Monge-Ampère
measure (ω+ ddcϕ)n is well defined, and there exists an increasing function
χ : R− → R− such that χ(−∞) = −∞ and χ ◦ ϕ ∈ L1((ω + ddcϕ)n).
In general χ has very slow growth at infinity, so that ϕ 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 ϕ ∈ E(X,ω) such that µ = (ω+ddcϕ)n if and
only if µ does not charge pluripolar sets.
This results has been established in [GZ 2] when ω is a Kähler form. It
is important for applications to complex dynamics and Kähler geometry to
consider as well forms ω that are less positive (see [EGZ]).
http://arxiv.org/abs/0704.0866v2
2 S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI
We then look for conditions on the measure µ which insure that the
solution ϕ is almost bounded. Following the seminal work of S. Kolodziej
[K 2,3], we say that µ is dominated by the Monge-Ampère Capacity Capω
if there exists a function F : R+ → R+ such that limt→0+ F (t) = 0 and
(†) µ(K) ≤ F (Capω(K)), for all Borel subsets K ⊂ X.
Here Capω denotes the global version of the Monge-Ampère capacity intro-
duced by E.Bedford and A.Taylor [BT] (see section 2).
Observe that µ does not charge pluripolar sets since F (0) = 0. When
F (x) . xα vanishes at order α > 1 and ω is Kähler, S. Kolodziej has
proved [K 2] that the solution ϕ ∈ PSH(X,ω) of (MA)µ is continuous. The
boundedness part of this result was extended in [EGZ] to the case when
ω is merely big and nonnegative. If F (x) . xα with 0 < α < 1, two of
us have proved in [GZ 2] that the solution ϕ has finite χ−energy, where
χ(t) = −(−t)p, p = p(α) > 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 Fε(x) = x[ε(− ln(x)/n)]
n where ε : R → [0,∞[ is nonincreasing.
Our second main result is:
THEOREM B. If µ(K) ≤ Fε(Capω(K)) for all Borel subsets K ⊂ X,
then µ = (ω + ddcϕ)n where ϕ ∈ PSH(X,ω) satisfies supX ϕ = 0 and
Capω(ϕ < −s) ≤ exp(−nH
−1(s)).
Here H−1 is the reciprocal function of H(x) = e
ε(t)dt + s0, where s0 =
s0(ε, ω) ≥ 0 only depends on ε and ω.
This general statement has several useful consequences:
ε(t)dt < +∞, thenH−1(s) = +∞ for s ≥ s∞ := e
ε(t)dt+
s0, hence Capω(ϕ < −s) = 0. This means that ϕ is bounded from
below by −s∞. This result is due to S. Kolodziej [K 2,3] when ω is
Kähler, and [EGZ] when ω ≥ 0 is merely big;
• the condition (†) is easy to check for measures with density in Lp, p >
1. Our result thus gives a simple proof (Corollary 3.2), following the
seminal approach of S. Kolodziej ([K2]), of the C0-a priori estimate
of S.T. Yau [Y], which is crucial for proving the Calabi conjecture
(see [T] for an overview);
• when
ε(t)dt = +∞, the solution ϕ is generally unbounded. The
faster ε(t) decreases towards zero, the faster the growth of H−1 at
infinity, hence the closer is ϕ from being bounded;
• the special case ε ≡ 1 is of particular interest. Here µ(·) ≤ Capω(·),
and our result shows that Capω(ϕ < −s) decreases exponentially
fast, hence ϕ has “ loglog-singularities”. These are the type of sin-
gularities of the metrics used in Arakelov geometry in relation with
measures µ = 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
A PRIORI ESTIMATES FOR SOLUTIONS OF MONGE-AMPÈRE EQUATIONS 3
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 ωn, and their density do not belong to Lp, for any p > 1.
2. Weakly singular quasiplurisubharmonic functions
The class E(X,ω) of ω-psh functions with finite weighted Monge-Ampère
energy has been introduced and studied in [GZ 2]. It is the largest subclass
of PSH(X,ω) on which the complex Monge-Ampère operator (ω+ddc·)n is
well-defined and the comparison principle is valid. Recall that ϕ ∈ E(X,ω)
if and only if (ω + ddcϕj)
n(ϕ ≤ −j) → 0, where ϕj := max(ϕ,−j).
2.1. The range of the Monge-Ampère operator. The range of the
operator (ω + ddc·)n acting on E(X,ω) has been characterized in [GZ 2]
when ω is a Kähler form. We extend here this result to the case when ω is
merely nonnegative and big.
Theorem 2.1. Assume ω is a smooth closed nonnegative (1,1) form on X,
and µ is a positive Radon measure such that µ(X) =
ωn > 0.
Then there exists ϕ ∈ E(X,ω) such that µ = (ω + ddcϕ)n if and only if µ
does not charge pluripolar sets.
Proof. We can assume without loss of generality that µ and ω are normalized
so that µ(X) =
ωn = 1. Consider, for A > 0,
CA(ω) := {ν probability measure / ν(K) ≤ A · Capω(K), for all K ⊂ X},
where Capω denotes the Monge-Ampère capacity introduced by E.Bedford
and A.Taylor in [BT] (see [GZ 1] for this compact setting). Recall that
Capω(K) := sup
(ω + ddcu)n / u ∈ PSH(X,ω), 0 ≤ u ≤ 1
We first show that a measure ν ∈ CA(ω) is the Monge-Ampère of a func-
tion ψ ∈ Ep(X,ω), for any 0 < p < 1, where
Ep(X,ω) := {ψ ∈ E(X,ω) / ψ ∈ Lp
(ω + ddcψ)n
Indeed, fix ν ∈ CA(ω), 0 < p < 1, and ωj := ω + εjΩ, where Ω is
a kähler form on X, and εj > 0 decreases towards zero. Observe that
PSH(X,ω) ⊂ PSH(X,ωj), hence Capω(.) ≤ Capωj(.), so that ν ∈ CA(ωj).
It follows from Proposition 3.6 and 2.7 in [GZ 1] that there exists C0 > 0
such that for any v ∈ PSH(X,ωj) normalized by supX v = −1, we have
Capωj(v < −t) ≤
, for all t ≥ 1.
This yields Ep(X,ωj) ⊂ L
p(ν): if v ∈ Ep(X,ωj) with supX v = −1, then
(−v)pdν = p ·
tp−1ν(v < −t)dt
≤ pA ·
tp−1Capω(v < −t)dt+ Cp
+ Cp < +∞.
4 S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI
It follows therefore from Theorem 4.2 in [GZ 2] that there exists ϕj ∈
Ep(X,ωj) with supX ϕj = −1 and (ωj+dd
n = cj ·ν, where cj =
ωnj ≥
1 decreases towards 1 as εj decreases towards zero. We can assume without
loss of generality that 1 ≤ cj ≤ 2. Observe that the ϕj ’s have uniformly
bounded energies, namely
(−ϕj)
p(ωj + dd
n ≤ 2
(−ϕj)
pdν ≤ 2
Since supX ϕj = −1, we can assume (after extracting a convergent subse-
quence) that ϕj → ϕ in L
1(X), where ϕ ∈ PSH(X,ω), supX ϕ = −1.
Set φj := (supl≥j ϕl)
∗. Thus φj ∈ PSH(X,ωj), and φj decreases towards
ϕ. Since φj ≥ ϕj , it follows from the “fundamental inequality” (Lemma 2.3
in [GZ 2]) that
(−φj)
p(ωj + dd
n ≤ 2n
(−ϕj)
p(ωj + dd
n ≤ C ′ < +∞.
Hence it follows from stability properties of the class Ep(X,ω) that ϕ ∈
Ep(X,ω) (see Proposition 5.6 in [GZ 2]). Moreover
(ωj + dd
n ≥ inf
(ωl + dd
n ≥ ν,
hence (ω + ddcϕ)n = lim(ωj + dd
n ≥ ν. Since
ωn = ν(X) = 1, this
yields ν = (ω + ddcϕ)n as claimed above.
We can now prove the statement of the theorem. One implication is
obvious: if µ = (ω+ddcϕ)n, ϕ ∈ E(X,ω), then µ does not charge pluripolar
sets, as follows from Theorem 1.3 in [GZ 2].
So we assume now µ that does not charge pluripolar sets. Since C1(ω) is
a compact convex set of probability measures which contains all measures
(ω + ddcu)n, u ∈ PSH(X,ω), 0 ≤ u ≤ 1, we can project µ onto C1(ω) and
get, by a generalization of Radon-Nikodym theorem (see [R], [Ce]),
µ = f · ν, ν ∈ C1(ω), 0 ≤ f ∈ L
1(ν).
Now ν = (ω + ddcψ)n for some ψ ∈ E1/2(X,ω), ψ ≤ 0, as follows from the
discussion above. Replacing ψ by eψ shows that we can actually assume ψ
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 µ = (ω+ddcϕ)n
for some ϕ ∈ E(X,ω). �
2.2. High energy and capacity estimates. Given χ : R− → R− an
increasing function, we consider, following [GZ 2],
Eχ(X,ω) :=
ϕ ∈ E(X,ω) /
(−χ)(−|ϕ|) (ω + ddcϕ)n < +∞
Alternatively a function ϕ ≤ 0 belongs to Eχ(X,ω) if and only if
(−χ) ◦ ϕj (ω + dd
n < +∞, where ϕj := max(ϕ,−j)
is the canonical approximation of ϕ by bounded ω-psh functions. When
χ(t) = −(−t)p, Eχ(X,ω) is the class E
p(X,ω) used in previous section.
The properties of classes Eχ(X,ω) are quite different whether the weight
χ is convex (slow growth at infinity) or concave. In previous works [GZ 2],
A PRIORI ESTIMATES FOR SOLUTIONS OF MONGE-AMPÈRE EQUATIONS 5
two of us were mainly interested in weights χ of moderate growth at infinity
(at most polynomial). Our main objective in the sequel is to construct
solutions ϕ of (MA)µ which are “almost bounded”, i.e. in classes Eχ(X,ω)
for concave weights χ of arbitrarily high growth.
For this purpose it is useful to relate the property ϕ ∈ Eχ(X,ω) to the
speed of decreasing of Capω(ϕ < −t), as t → +∞. We set
Êχ(X,ω) :=
ϕ ∈ PSH(X,ω) /
tnχ′(−t)Capω(ϕ < −t)dt < +∞
An important tool in the study of classes Eχ(X,ω) are the “fundamental
inequalities” (Lemmas 2.3 and 3.5 in [GZ 2]), which allow to compare the
weighted energy of two ω-psh functions ϕ ≤ ψ. These inequalities are only
valid for weights of slow growth (at most polynomial), while they become
immediate for classes Êχ(X,ω). So are the convexity properties of Êχ(X,ω).
We summarize this and compare these classes in the following:
Proposition 2.2. The classes Êχ(X,ω) are convex and stable under maxi-
mum: if Êχ(X,ω) ∋ ϕ ≤ ψ ∈ PSH(X,ω), then ψ ∈ Êχ(X,ω).
One always has Êχ(X,ω) ⊂ Eχ(X,ω), while
Eχ̂(X,ω) ⊂ Êχ(X,ω), where χ
′(t− 1) = tnχ̂′(t).
Since we are mainly interested in the sequel in weights with (super)
fast growth at infinity, the previous proposition shows that Êχ(X,ω) and
Eχ(X,ω) are roughly the same: a function ϕ ∈ PSH(X,ω) belongs to one of
these classes if and only if Capω(ϕ < −t) decreases fast enough, as t→ +∞.
Proof. The convexity of Êχ(X,ω) follows from the following simple observa-
tion: if ϕ,ψ ∈ Êχ(X,ω) and 0 ≤ a ≤ 1, then
{aϕ+ (1− a)ψ < −t} ⊂ {ϕ < −t} ∪ {ψ < −t} .
The stability under maximum is obvious.
Assume ϕ ∈ Êχ(X,ω). We can assume without loss of generality ϕ ≤ 0
and χ(0) = 0. Set ϕj := max(ϕ,−j). It follows from Lemma 2.3 below that
(−χ) ◦ ϕj (ω + dd
χ′(−t)(ω + ddcϕj)
n(ϕj < −t)dt
χ′(−t)tnCapω(ϕ < −t)dt < +∞,
This shows that ϕ ∈ Eχ(X,ω). The other inclusion goes similarly, using
the second inequality in Lemma 2.3 below. �
If ϕ ∈ Eχ(X,ω) (or Êχ(X,ω)), then the bigger the growth of χ at −∞,
the smaller Capω(ϕ < −t) when t → +∞, hence the closer ϕ is from being
bounded. Indeed ϕ ∈ PSH(X,ω) is bounded iff it belongs to Eχ(X,ω) for
all weights χ, as was observed in [GZ 2], Proposition 3.1. Similarly
PSH(X,ω) ∩ L∞(X) =
Êχ(X,ω),
where the intersection runs over all concave increasing functions χ.
We will make constant use of the following result:
6 S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI
Lemma 2.3. Fix ϕ ∈ E(X,ω). Then for all s > 0 and 0 ≤ t ≤ 1,
tnCapω(ϕ < −s− t) ≤
(ϕ<−s)
(ω + ddcϕ)n ≤ snCapω(ϕ < −s),
where the second inequality is true only for s ≥ 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 µ denotes a positive Radon measure on X whose total mass
is V olω(X): this is an obvious necessary condition in order to solve (MA)µ.
To simplify numerical computations, we assume in the sequel that µ and ω
have been normalized so that
µ(X) = V olω(X) =
ωn = 1.
When µ = ehωn is a smooth volume form and ω is a Kähler form, S.T.Yau
has proved [Y] that (MA)µ admits a unique smooth solution ϕ ∈ PSH(X,ω)
with supX ϕ = 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 exten-
sively studied by S.Kolodziej in [K 2,3,4]. Following S. Kolodziej ([K3], [K4])
with slightly different notations, fix ε : R → [0,∞[ a continuous decreasing
function and set
Fε(x) := x[ε(− lnx/n)]
n, x > 0.
We will consider probability measures µ satisfying the following condition :
for all Borel subsets K ⊂ X,
µ(K) ≤ Fε(Capω(K)).
The main result achieved in [K 2], can be formulated as follows: If ω is a
Kähler form and
ε(t)dt < +∞ then µ = (ω + ddcϕ)n for some contin-
uous function ϕ ∈ PSH(X,ω).
The condition
ε(t)dt < +∞ means that ε decreases fast enough
towards zero at infinity. This gives a quantitative estimate on how fast
ε(− lnCapω(K)/n), hence µ(K), decreases towards zero as Capω(K) → 0.
ε(t)dt = +∞, it follows from Theorem 2.1 that µ = (ω +
ddcϕ)n for some function ϕ ∈ E(X,ω), but ϕ will generally be unbounded.
Our second main result measures how far ϕ is from being bounded:
Theorem 3.1. Assume for all compact subsets K ⊂ X,
(3.1) µ(K) ≤ Fε(Capω(K)).
Then µ = (ω + ddcϕ)n where ϕ ∈ E(X,ω) is such that supX ϕ = 0 and
Capω(ϕ < −s) ≤ exp(−nH
−1(s)), for all s > 0.
Here H−1 is the reciprocal function of H(x) = e
ε(t)dt + s0, where s0 =
s0(ε, ω) ≥ 0 is a constant which only depends on ε and ω.
In particular ϕ ∈ Eχ(X,ω) where −χ(−t) = exp(nH
−1(t)/2).
A PRIORI ESTIMATES FOR SOLUTIONS OF MONGE-AMPÈRE EQUATIONS 7
Recall that here, and troughout the article, ω ≥ 0 is merely big.
Before proving this result we make a few observations.
• It is interesting to consider as well the case when ε(t) increases to-
wards +∞. One can then obtain solutions ϕ such that Capω(ϕ <
−t) decreases at a polynomial rate. When e.g. ω is Kähler and
µ(K) ≤ Capω(K)
α, 0 < α < 1, it follows from Proposition 5.3
in [GZ 2] that µ = (ω + ddcϕ)n where ϕ ∈ Ep(X,ω) for some
p = pα > 0. Here E
p(X,ω) denotes the Cegrell type class Eχ(X,ω),
with χ(t) = −(−t)p.
• When ε(t) ≡ 1, Fε(x) = x and H(x) ≍ e.x. Thus Theorem 3.1 reads
µ ≤ Capω ⇒ µ = (ω + dd
cϕ)n, where
Capω(ϕ < −s) . exp (−ns/e) .
This is precisely the rate of decreasing corresponding to functions
which look locally like − log(− log ||z||), in some local chart z ∈
U ⊂ Cn. This class of ω-psh functions with “loglog-singularities” is
important for applications (see [Ku], [BKK]).
• If ε(t) decreases towards zero, then Capω(ϕ < −t) decreases at a
superexponential rate. The faster ε(t) decreases towards zero, the
slower the growth of H, hence the faster the growth of H−1 at infin-
ity. When
ε(t)dt < +∞, the function ε decreases so fast that
Capω(ϕ < −t) = 0 for t >> 1, thus ϕ is bounded. This is the case
when µ(K) ≤ Capω(K)
α for some α > 1 [K 2], [EGZ].
• When
ε(t)dt = +∞, the solution ϕmay well be unbounded (see
Examples in section 4). At the critical case where µ ≤ Fε(Capω) for
all functions ε such that
ε(t)dt = +∞, we obtain
µ = (ω + ddcϕ)n with ϕ ∈ PSH(X,ω) ∩ L∞(X),
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 µ implies in particular that it vanishes on pluripo-
lar sets. It follows from Theorem 2.1 that there exists a function ϕ ∈ E(X,ω)
such that µ = (ω + ddcϕ)n and supX ϕ = 0. Set
g(s) := −
logCapω(ϕ < −s), ∀s > 0.
The function g is increasing on [0,+∞] and g(+∞) = +∞, since Capω
vanishes on pluripolar sets. Observe also that g(s) ≥ 0 for all s ≥ 0, since
g(0) = −
logCapω(X) = −
log V olω(X) = 0.
It follows from Lemma 2.3 and (3.1) that for all s > 0 and 0 ≤ t ≤ 1,
tnCapω(ϕ < −s− t) ≤ µ(ϕ < −s) ≤ Fε (Capω(ϕ < −s)) .
Therefore for all s > 0 and 0 ≤ t ≤ 1,
(3.2) log t− log ε ◦ g(s) + g(s) ≤ g(s + t).
We define an increasing sequence (sj)j∈N by induction setting
sj+1 = sj + eε ◦ g(sj), for all j ∈ N.
8 S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI
The choice of s0. Recall that (3.2) is only valid for 0 ≤ t ≤ 1. We choose
s0 ≥ 0 large enough so that
(3.3) e.ε ◦ g(s0) ≤ 1.
This will allow us to use (3.2) with t = tj = sj+1 − sj ∈ [0, 1], since ε ◦ g is
decreasing, while sj ≥ s0 is increasing, hence
0 ≤ tj = eε ◦ g(sj) ≤ eε ◦ g(s0) ≤ 1.
We must insure that s0 = s0(ε, ω) can chosen to be independent of ϕ. This
is a consequence of Proposition 2.7 in [GZ 1]: since supX ϕ = 0, there exists
c1(ω) > 0 so that 0 ≤
(−ϕ)ωn ≤ c1(ω), hence
g(s) := −
logCapω(ϕ < −s) ≥
log s−
log(n+ c1(ω)).
Therefore g(s0) ≥ ε
−1(1/e) for s0 = s0(ε, ω) := (n+ c1(ω)) exp(nε
−1(1/e)),
which is independent of ϕ. This yields e.ε ◦ g(s0) ≤ 1, as desired.
The growth of sj. We can now apply (3.2) and get g(sj) ≥ j + g(s0) ≥ j.
Thus lim g(sj) = +∞. There are two cases to be considered.
If s∞ = lim sj ∈ R
+, then g(s) ≡ +∞ for s > s∞, i.e. Capω(ϕ < −s) =
0, ∀s > s∞. Therefore ϕ is bounded from below by −s∞, in particular
ϕ ∈ Eχ(X,ω) for all χ.
Assume now (second case) that sj → +∞. For each s > 0, there exists
N = Ns ∈ N such that sN ≤ s < sN+1. We can estimate s 7→ Ns:
s ≤ sN+1 =
(sj+1 − sj) + s0 =
e ε ◦ g(sj) + s0
ε(j) + s0 ≤ e.ε(0) + e
ε(t)dt+ s0 =: H(N),
Therefore H−1(s) ≤ N ≤ g(sN ) ≤ g(s), hence
Capω(ϕ < −s) ≤ exp(−nH
−1(s)).
Set now −χ(−t) = exp(nH−1(t)/2). Then
tnχ′(−t)Capω(ϕ < −t)dt
ε(H−1(t)) + s̃0
exp(−nH−1(t)/2)dt
tn exp(−nt/2)dt < +∞.
This shows that ϕ ∈ Eχ(X,ω) where χ(t) = − exp(nH
−1(−t)/2).
It follows from the proof above that when
ε(t)dt < +∞, the solution
ϕ is bounded since in this case we have
s∞ := lim
sj ≤ s0(ε, ω) + e ε(0) + e
ε(t)dt < +∞
where s0(ε, ω) is an absolute constant satisfying (3.3) (see above). �
A PRIORI ESTIMATES FOR SOLUTIONS OF MONGE-AMPÈRE EQUATIONS 9
Let us emphasize that Theorem 3.1 also yields a slightly simplified proof of
the following result [K 2], [EGZ]: if µ(K) ≤ Fε(Capω(K)) for some decreas-
ing function ε : R → R+ such that
ε(t)dt < +∞, then the sequence
(sj) above is convergent, hence µ = (ω + dd
cϕ)n, where ϕ ∈ PSH(X,ω) is
bounded. For the reader’s convenience we indicate a proof of the following
important particular case:
Corollary 3.2. Let µ = fωn be a measure with density 0 ≤ f ∈ Lp(ωn),
where p > 1 and
fωn =
ωn. Then there exists a unique bounded
function ϕ ∈ PSH(X,ω) such that (ω + ddcϕ)n = µ, supX ϕ = 0 and
0 ≤ ||ϕ||L∞(X) ≤ C(p, ω).||f ||
Lp(ωn)
where C(p, ω) > 0 only depends on p and ω.
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 ω 7−→ C(p, ω) is quite explicit, as we shall see in the
proof. This is important when considering degenerate situations [EGZ].
Proof. We claim that there exists C1(ω) such that
(3.4) µ(K) ≤
C1(ω)||f ||
Lp(ωn)
[Capω(K)]
, for all Borel sets K ⊂ X.
Assuming this for the moment, we can apply Theorem 3.1 with ε(x) =
C1(ω)||f ||
Lp(ωn)
exp(−x), which yields, as observed at the end of the proof
of Theorem 3.1
||ϕ||L∞(X) ≤M(f, ω),
whereM(f, ω) := s0(ε, ω)+e ε(0)+e
ε(t)dt = s0(ε, ω)+2eC1(ω)||f ||
Lp(ωn)
and s0 = s0(ε, ω) is a large number s0 > 1 satisfying the inequality (3.3).
In order to give the precise dependence of the uniform bound M(f, ω) on
the Lp−norm of the density f , we need to choose s0 more carefully. Observe
that condition (3.3) can be written
Capω({ϕ ≤ −s0}) ≤ exp(−nε
−1(1/e).
Since nε−1(1/e) = log
enC1(ω)
n‖f‖Lp(ωn)
, we must choose s0 > 0 so that
(3.5) Capω({ϕ ≤ −s0}) ≤
enC1(ω)n‖f‖Lp(ωn)
We claim that for any N ≥ 1 there exists a uniform constant C2(N, p, ω) >
0 such that for any s > 0,
(3.6) Capω({ϕ ≤ −s}) ≤ C2(N, p) s
−N ‖f‖Lp(ωn).
Indeed observe first that by Hölder inequality,
(−ϕ)Nωnϕ =
(−ϕ)Nfωn ≤ ‖f‖Lp(ωn)‖ϕ‖
LNq (ωn)
10 S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI
Since ϕ belongs to the compact family {ψ ∈ PSH(X,ω); supX ψ = 0}
([GZ2]), there exists a uniform constant C ′2(N, p, ω) > 0 such that ‖ϕ‖
LNq(ωn)
C ′2(N, p, ω), hence
(−ϕ)Nωnϕ ≤ C
2(N, p, ω)‖f‖Lp(ωn).
Fix u ∈ PSH(X,ω) with −1 ≤ u ≤ 0 and N ≥ 1 to be specified later. If
follows from Tchebysheff and energy inequalities ([GZ2]) that
{ϕ≤−s}
(ω + ddcu)n ≤ s−N
(−ϕ)N (ω + ddcu)n
≤ cN s
−N max
(−ϕ)Nωnϕ,
(−u)Nωnu
≤ cN s
−N max
C ′2(N, p, ω), 1
‖f‖Lp(ωn).
We have used here the fact that ‖f‖Lp(ωn) ≥ 1, which follows from the
normalization : 1 =
fωn ≤ ‖f‖Lp(ωn). This proves the claim.
SetN = 2n, it follows from (3.6) that s0 := C1(ω)
nenC2(2n, p, ω)‖f‖
Lp(ωn)
satisfies the required condition (3.5), which implies the estimate of the the-
orem.
We now establish the estimate (3.4). Observe first that Hölder’s inequality
yields
(3.7) µ(K) ≤ ||f ||Lp(ωn) [V olω(K)]
, where 1/p + 1/q = 1.
Thus it suffices to estimate the volume V olω(K). Recall the definition of
the Alexander-Taylor capacity, Tω(K) := exp(− supX VK,ω), where
VK,ω(x) := sup{ψ(x) /ψ ∈ PSH(X,ω), ψ ≤ 0 on K}.
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):
(3.8) Tω(K) ≤ e exp
Capω(K)
It thus remains to show that V olω(K) is suitably bounded from above by
Tω(K). This follows from Skoda’s uniform integrability result: set
ν(ω) := sup {ν(ψ, x) /ψ ∈ PSH(X,ω), x ∈ X} ,
where ν(ψ, x) denotes the Lelong number of ψ at point x. This actually
only depends on the cohomology class {ω} ∈ H1,1(X,R). It is a standard
fact that goes back to H.Skoda (see [Z]) that there exists C2(ω) > 0 so that
ωn ≤ C2(ω),
for all functions ψ ∈ PSH(X,ω) normalized by supX ψ = 0. We infer
(3.9) V olω(K) ≤
V ∗K,ω
ωn ≤ C2(ω)[Tω(K)]
1/ν(ω).
A PRIORI ESTIMATES FOR SOLUTIONS OF MONGE-AMPÈRE EQUATIONS 11
It now follows from (3.7), (3.8), (3.9), that
µ(K) ≤ ||f ||Lp [C2(ω)]
1/qe1/qν(ω) exp
qν(ω)Capω(K)
The conclusion follows by observing that exp(−1/x1/n) ≤ Cnx
2 for some
explicit constant Cn > 0. �
4. Examples
4.1. Measures invariant by rotations. In this section we produce exam-
ples 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 = P1(C), with
ω = ωFS, the Fubini-Study volume form. Consider µ = fω a measure with
density f which is smooth and positive on X \ {p}, and such that
f(z) ≃
|z|2(log |z|)2
, c > 0,
in a local chart near p = 0. A simple computation yields µ = ω + ddcϕ,
where ϕ ∈ PSH(P1, ω) is smooth in P1 \ {p} and ϕ(z) ≃ −c′ log(− log |z|)
near p = 0, c′ > 0, hence
logCapω(ϕ < −t) ≃ −t,
Here a ≃ b means that a/b is bounded away from zero and infinity.
This is to be compared to our estimate logCapω(ϕ < −t) . −t/e (Theo-
rem 3.1 ) which can be applied, as it was shown by S.Kolodziej in [K 1] that
µ . Capω. Thus Theorem 3.1 is essentially sharp when ε ≡ 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 ε as in Theorem 3.1. Consider µ = fω on X = P1(C),
where ω = ωFS is the Fubini-Study volume form, f ≥ 0 is continuous on
1 \ {p}, and
f(z) ≃
ε(log(− log |z|))
|z|2(log |z|)2
in local coordinates near p = 0. Here ε : R → R+ decreases towards 0 at
+∞. We claim that there exists A > 0 such that
(4.1) µ(K) ≤ ACapω(K)ε(− logCapω(K)), for all K ⊂ X.
This is clear outside a small neighborhood of p = 0 since the measure µ
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
K̃ := {r ∈ [0, R] ; K ∩ {|z| = r} 6= ∅}.
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 K̃, namely
l(K̃)
≤ c(K̃) ≤ c(K).
12 S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI
Using that ε is decreasing, hence 0 ≤ −ε′, we infer
µ(K) ≤ 2π
∫ l(K̃)
f(r)rdr
∫ l(K̃)
ε(log(− log r))− ε′(log− log r)
r(log r)2
ε(log(− log l(K̃)))
− log l(K̃)
ε(log(− log 4c(K)))
− log 4c(K)
Recall now that the logarithmic capacity c(K) is equivalent to Alexander-
Taylor’s capacity T∆(K), which in turn is equivalent to the global Alexander-
Taylor capacity Tω(K) (see [GZ 1]): c(K) ≃ T∆(K) ≃ Tω(K). The Alexander-
Taylor’s comparison theorem [AT] reads
− log 4c(K) ≃ − log Tω(K) ≃ 1/Capω(K),
thus µ(K) ≤ ACapω(K)ε(− logCapω(K)).
We can therefore apply Theorem 3.1. It guarantees that µ = (ω + ddcϕ),
where ϕ ∈ PSH(P1, ω) satisfies logCapω(ϕ < −s) ≃ −nH
−1(s), with
H(s) = eA
ε(t)dt + s0. On the other hand a simple computation shows
that ϕ is continuous in P1 \ {p} and
ϕ ≃ −H(log(− log |z|)) , near p = 0.
The sublevel set (ϕ < −t) therefore coincides with the ball of radius
exp(− exp(H−1(t))), hence logCapω(ϕ < −s) ≃ −H
−1(s).
4.2. Measures with density. Here we consider the case when µ = fdV
is absolutely continuous with respect to a volume form.
Proposition 4.3. Assume µ = fωn is a probability measure whose density
satisfies f [log(1 + f)]n ∈ L1(ωn). Then µ . Capω.
More generally if f [log(1 + f)/ε(log(1 + | log f |))]n ∈ L1(ωn) for some
continuous decreasing function ε : R → R+∗ , then for all K ⊂ X,
µ(K) ≤ Fε(Capω(K)), where Fε(x) = Ax
, A > 0.
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 ϕ(z) = − log(− log ||z||), where ||z|| =
|z1|2 + . . .+ |zn|2 de-
notes the Euclidean norm in Cn. One can check that ϕ is plurisubharmonic
in a neighborhood of the origin in Cn, and that there exists cn > 0 so that
µ := (ddcϕ)n = f dVeucl, where f(z) =
||z||2n(− log ||z||)n+1
Observe that f [log(1 + f)]n−α ∈ L1, ∀α > 0 but f [log(1 + f)]n 6∈ L1.
A PRIORI ESTIMATES FOR SOLUTIONS OF MONGE-AMPÈRE EQUATIONS 13
When n = 1 it was observed by S. Kolodziej [K 1] that µ(K) . Capω(K).
Proposition 4.3 yields here
µ(K) . Capω(K)(| logCapω(K)|+ 1).
For n ≥ 1, it follows from Proposition 4.3 and Theorem 3.1 that
logCapω(ϕ < −s) . −nH
−1(s).
On the other hand, one can directly check that logCapω(ϕ < −s) ≃ −nH
−1(s).
One can get further examples by considering ϕ(z) = χ ◦ log ||z||, so that
(ddcϕ)n =
′ ◦ log ||z||)n−1χ′′(log ||z||)
||z||2n
dVeucl.
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http://arxiv.org/abs/math/0603431
14 S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI
Slimane BENELKOURCHI & Vincent GUEDJ & Ahmed ZERIAHI
Laboratoire Emile Picard
UMR 5580, Université Paul Sabatier
118 route de Narbonne
31062 TOULOUSE Cedex 09 (FRANCE)
benel@math.ups-tlse.fr
guedj@math.ups-tlse.fr
zeriahi@math.ups-tlse.fr
	1. Introduction
	2. Weakly singular quasiplurisubharmonic functions
	2.1. The range of the Monge-Ampère operator
	2.2. High energy and capacity estimates
	3. Measures dominated by capacity
	4. Examples
	4.1. Measures invariant by rotations
	4.2. Measures with density
	References
	Bibliography
