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Journal of Functional Analysis ••• (••••) •••–•••www.elsevier.com/locate/jfa

Characterization of Monge–Ampère measures withHölder continuous potentials

Tien-Cuong Dinh a,b, Viêt-Anh Nguyên c,∗

a UPMC Univ Paris 06, UMR 7586, Institut de Mathématiques de Jussieu, 4 place Jussieu, F-75005 Paris, Franceb DMA, UMR 8553, Ecole Normale Supérieure, 45 rue d’Ulm, 75005 Paris, France

c Mathématique-Bâtiment 425, UMR 8628, Université Paris-Sud, 91405 Orsay, France

Received 10 June 2012; accepted 27 August 2013

Communicated by I. Rodnianski

Abstract

We show that the complex Monge–Ampère equation on a compact Kähler manifold (X,ω) of dimen-sion n admits a Hölder continuous ω-psh solution if and only if its right-hand side is a positive measurewith Hölder continuous super-potential. This property is true in particular when the measure has locallyHölder continuous potentials or when it belongs to the Sobolev space W2n/p−2+ε,p(X) or to the Besovspace Bε−2∞,∞(X) for some ε > 0 and p > 1.© 2013 Elsevier Inc. All rights reserved.

Keywords: complex Monge–Ampère equation; p.s.h. function; super-potential; capacity; moderate measure

1. Introduction

Let X be a compact Kähler manifold of dimension n and ω a Kähler form on X. Let d, dc

denote the real differential operators on X defined by d := ∂ + ∂ , dc := 12πi

(∂ − ∂) so thatddc = i

π∂∂ . Recall that a function u : X → R ∪ {−∞} is said to be ω-psh if it is locally the

* Corresponding author.E-mail addresses: dinh@math.jussieu.fr (T.-C. Dinh), VietAnh.Nguyen@math.u-psud.fr (V.-A. Nguyên).URLs: http://www.math.jussieu.fr/~dinh (T.-C. Dinh), http://www.math.u-psud.fr/~vietanh (V.-A. Nguyên).

0022-1236/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jfa.2013.08.026

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difference of a p.s.h. function and a potential of ω, i.e., a locally defined smooth function v suchthat ddcv = ω.

We consider the complex Monge–Ampère equation

(ddcϕ + ω

)n = μ,

where μ is a positive measure on X and ϕ a bounded ω-psh function, see Bedford and Taylor [3],Demailly [7] and Fornæss and Sibony [19] for the intersection of currents and for basic propertiesof p.s.h. functions. For cohomology reason, the above relation implies that the mass of μ, i.e.,‖μ‖ := μ(X), is equal to the mass of the measure induced by the volume form ωn, that is,

‖μ‖ =∫X

ωn.

In what follows, we always assume this condition.When μ is given by a smooth volume form, the famous theorem of Calabi–Yau says that

the Monge–Ampère equation admits a smooth solution ϕ which is unique up to an additiveconstant [5,31]. When μ only satisfies weak assumptions of regularity, Hölder continuity is thebest global regularity one can expect. Indeed, without reviewing the long history of the complexMonge–Ampère equation, let us mention few steps in the recent developments.

In [24,25], Kolodziej has constructed a continuous solution under some hypothesis on themeasure μ, in particular for μ of class Lp , p > 1. Then, he proved in [26] that the solutionis Hölder continuous when μ is of class Lp , p > 1, see also [18,20]. Some important steps inhis approach were improved by Dinew and Zhang [11]. In particular, these authors obtained aversion of Kolodziej’s theorem on stability of the solution in the case where μ is not absolutelycontinuous with respect to the Lebesgue measure.

Very recently based on Demailly’s regularization method [6] and the above Dinew–Zhang’sresults, Demailly, Dinew, Guedj, Hiep, Kolodziej, Zeriahi [8] obtained an explicit Hölder ex-ponent of the solution, see also [10]. A necessary condition on μ to have a Hölder continuoussolution u was obtained by Dinh, Nguyen and Sibony in [12]. We refer to the works of theabove mentioned authors and Eyssidieux, Pali, Plis, Tian [8–12,18,23–27,29], for results in thisdirection, for related topics and a more complete list of references.

The main purpose of this work is to investigate the complex Monge–Ampère equation withHölder continuous solutions using the theory of super-potentials. The notion of super-potentialwas introduced by Sibony and the first author in [16,17]. We recall it in the case of measures,i.e., for p = n = dim(X).

Let C denote the set of positive closed (1,1)-currents in the cohomology class {ω}. This is aconvex compact set. For each current T in C there is a unique ω-psh function u such that

T = ddcu + ω and∫X

uωn = 0.

We call u the ω-potential of T . Define the following distance on C

distL1

(T ,T ′) := ∥∥u − u′∥∥

1 ,

L

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where u and u′ are ω-potentials of T and T ′ respectively and the L1-norm is with respect to themeasure induced by ωn.

Definition 1.1. A function U : C →R is said to be Hölder continuous if it is Hölder continuouswith respect to the distance distL1 on C .

Definition 1.2. Let μ be a positive measure on X. The super-potential of μ is the functionU : C → R∪ {−∞} defined by

U (T ) :=∫

udμ

where u is the ω-potential of T .

It follows from this definition that if μ is given by a bounded differential form then its super-potential is Lipschitz with respect to the distance distL1 on C .

The infinite dimensional compact space C admits some “complex structure” and the super-potential U satisfies similar properties as the ones of quasi-psh functions in the finite dimen-sional case, see [16,17]. However, we do not need these properties here. It is important to noticethat the Hölder continuity, the continuity and the boundedness of the super-potential U do notdepend on the choice of the Kähler form ω on X.

Super-potentials were introduced for positive closed currents of all bidegree. Up to somenatural normalization, the super-potential of a current is a canonical function associated to thiscurrent and plays the same role as ω-potentials in the case of bidegree (1,1). Several applicationshave been obtained in complex dynamics where the role of super-potentials is crucial. Here isour main theorem which is also the first application of super-potentials in the Monge–Ampèreequation. The proof uses the above mentioned results and some ideas from the works by Sibonyand the authors [12,13,16,17].

Theorem 1.3. Let (X,ω) be a compact Kähler manifold of dimension n. Let μ be a positivemeasure on X of mass

∫X

ωn. Then the Monge–Ampère equation

(ddcϕ + ω

)n = μ

admits a Hölder continuous ω-psh solution ϕ if and only if the super-potential of μ defined aboveis Hölder continuous.

The Hölder exponent of the solution depends on n and on the Hölder exponent of the super-potential U of μ. It will be specified in the proof of Theorem 1.3 in Section 3. In the samesection, we will give some properties of super-potentials for measures.

In Refs. [16,17], situations where the Hölder continuity of super-potential can be verified,are described. The reader will also find in Section 4 some methods to obtain this property. Inparticular, we will see that the class of measures with Hölder continuous super-potential containsthe measures given by Lp densities, p > 1, considered in [8,11,26] and the measures satisfyingsome Hausdorff type regularity investigated in [23]. Section 5 is devoted to the final step of theproof of Theorem 1.3.

The necessary condition in this theorem is obtained in Proposition 4.1. The rough idea is thatwhen ϕ is of class C 2 then μ is a bounded differential form. It follows from the discussion after

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Definition 1.2 that the super-potential of μ is Lipschitz. For the general case with a Hölder con-tinuous function ϕ, we need to approximate it by suitable C 2 functions and appropriate estimatesgive the result. For the sufficient condition, assuming the Hölder continuity of the super-potentialof μ, we first obtain a good comparison between μ and a notion of capacity on X which, accord-ing to [24], allows us to obtain a bounded solution ϕ of the considered Monge–Ampère equation.Further estimates on certain regularizations of ϕ are then obtained. They permit us to adapt theapproach of [8] which gives the Hölder continuity of ϕ.

One may expect an analogue of Theorem 1.3 in the local setting, e.g. for the complex Monge–Ampère equation on a bounded smooth strictly pseudoconvex domain in C

n with the Dirichletcondition on the boundary. In order to define the super-potential of a measure, one can replace Cby a space of p.s.h. functions. However, there is here a new difficulty on the choice of this space,and more precisely, on the behavior of these p.s.h. functions near the boundary of the domain. Itis not clear that a common space for both necessary and sufficient conditions could exist in thissetting.

The following corollary is a consequence of Theorem 1.3 and Proposition 4.2 below.

Corollary 1.4. Under the hypotheses of Theorem 1.3, if we can write locally μ = ddcU + ∂V +∂W with Hölder continuous forms U,V,W of bidegrees (n− 1, n− 1), (n− 1, n) and (n,n− 1)

respectively, then the considered Monge–Ampère equation admits a Hölder continuous ω-pshsolution. Moreover, the hypothesis on μ is satisfied when this measure belongs to the Sobolevspace W 2n/p−2+ε,p(X) or to the Besov space Bε−2∞,∞(X) for some ε > 0 and p > 1.

The definition of the Besov space in the above corollary is technical. We will recall it justbefore Proposition 4.2 below.

In the case with parameters (Xt ,ωt ,μt ), where the compact Kähler manifolds (Xt ,ωt ) haveuniformly bounded geometry and the super-potentials of the measures μt are uniformly Höldercontinuous, we obtain solutions ϕt which are uniformly Hölder continuous. We can also extendour results to the case of a big and nef class with a solution locally Hölder continuous on theample locus. The proof requires some extra techniques that can be found in [8,11,18].

The following problem suggested by the work of Sibony and the authors [12] is still open. Werefer to [8,13] for some particular cases where the answer is positive.

Problem 1.5. Let μ be a probability measure on X. Assume that μ is moderate. Does the Monge–Ampère equation

(ddcϕ + ω

)n = μ

admit a Hölder continuous ω-psh solution ϕ?

The notion of moderate measures will be recalled in Section 2. We think that the answer isnot always positive and the problem requires probably a better understanding of the notion ofcapacity T (·), see [15] and Section 2 for the definition.

Problem 1.6. Characterize the positive measures μ on X such that the associated complexMonge–Ampère equation admits a continuous (resp. bounded) solution.

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Note that it is not difficult to show using [7, Ch. III (3.6) and (3.11)] that when ϕ is a continu-ous (resp. bounded) ω-psh function the measure (ddcϕ + ω)n has a continuous (resp. bounded)super-potential. However, in dimension n� 2, the last property does not characterize the Monge–Ampère measures with continuous (resp. bounded) potential, see Example 3.4 below.

Finally, for the reader’s convenience, we mention that the main references involved in thetechnical part of this work are [8,11,12,14,16,24].

2. Moderate measures and capacities

In what follows, (X,ω) always denotes a compact Kähler manifold of dimension n. Recallfrom [12,13] that a positive measure μ on X is moderate if there are constants c > 0 and α > 0such that ∫

e−αu dμ � c for every u ω-psh such that∫X

uωn = 0.

Note that the functions u satisfying the last condition describe a compact set of ω-psh func-tions. The condition can be replaced by other ones, e.g. maxX u = 0. In the case where μ is theLebesgue measure on X, the supremum of all α satisfying the above estimate is called Tian’sinvariant; it was introduced by Tian in [29].

Lemma 2.1. Let μ be a moderate positive measure on X. Then there are constants c > 0 andα > 0 such that for any M � 0

μ{u < −M}� ce−αM for every ω-psh function u such that∫X

uωn = 0.

Moreover, if p � 1 is a real number, then there is a constant cp > 0 such that

‖u‖Lp(μ) � cp and∫

{u<−M}|u|p dμ� cpe−αM for all M and u as above.

Proof. We have

μ{u < −M}� e−αM

∫{u<−M}

e−αu dμ� e−αM

∫e−αu dμ� ce−αM,

where the last estimate follows from the definition of moderate measures. This gives the firstassertion in the lemma.

Observe that u is bounded above by a constant independent of u. Therefore, the identities

∫ (|u|p − Mp)dμ =

∞∫p

μ{|u|p > r

}dr = p

∞∫μ

{|u| > t}tp−1 dt

|u|>M M M

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for M � 0, combined with the first assertion, implies that, for 0 < α′ < α there is a constantcp > 0 such that

∫{u<−M}

|u|p dμ � cpe−α′M.

for all M � 0. It is then enough to replace α by a smaller constant in order to get the secondassertion of the lemma. �

The following lemma gives a quantitative version of the property that bounded sets of ω-pshfunctions in L1(μ) are also bounded in Lp(μ) for all 1 � p < ∞. The property is well-knownfor the measures with bounded densities, see e.g. [7].

Lemma 2.2. Let μ be a moderate positive measure on X. Then for any real number p � 1, thereis a constant c > 0 such that

∥∥u − u′∥∥Lp(μ)

� c max(1,− log

∥∥u − u′∥∥L1(μ)

)(p−1)/p∥∥u − u′∥∥1/p

L1(μ)

for all ω-psh functions u and u′ such that∫X

uωn = ∫X

u′ωn = 0.

Proof. In what follows, � and � denote inequalities up to a multiplicative constant. Since weknow from Lemma 2.1 that ‖u‖Lp(μ) and ‖u′‖Lp(μ) are bounded, we only have to consider thecase where ‖u − u′‖L1(μ) is small. Define for a constant M large enough uM := max(u,−M)

and u′M := max(u′,−M). Using Lemma 2.1 and Hölder’s inequality, we have

∥∥u − u′∥∥Lp(μ)

�∥∥uM − u′

M

∥∥Lp(μ)

+ e−αM � M(p−1)/p∥∥uM − u′

M

∥∥1/p

L1(μ)+ e−αM

� M(p−1)/p∥∥u − u′∥∥1/p

L1(μ)+ M(p−1)/pe−αM/p + e−αM.

It is enough to choose M equal to a large constant times − log‖u − u′‖L1(μ). �Recall the following notion of capacity which was introduced by Kolodziej [25] and is related

to the well-known Bedford–Taylor capacity [3]. For any Borel subset A of X, define

capBTK(A) := sup

{∫A

(ddcu + ω

)n, u ω-psh such that 0 � u� 1

}.

The following definition is inspired by the works of Kolodziej [24–26].

Definition 2.3. A positive measure μ on X is said to be K-moderate if there are constants c > 0and α > 0 such that for every Borel subset A of X we have

μ(A) � c exp(− capBTK(A)−α

).

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We will also need the following notion of capacity introduced by Sibony and the first author[15] which is related to the capacities of Alexander [2] and of Siciak [28], see also Harvey andLawson [22, p. 616]. For any Borel subset A of X, define

T (A) := inf{

exp(

supA

u), u ω-psh and max

Xu = 0

}.

We have T (A) = 0 if and only if A is pluripolar, i.e., it is contained in the set {u = −∞} forsome ω-psh function u.

Recall from Guedj and Zeriahi [21, Prop. 7.1] the following relation between capBTK(A) andT (A) for every compact set A ⊂ X:

c1 exp(−λ1 capBTK(A)−1) � T (A) � c2 exp

(−λ2 capBTK(A)−1/n), (1)

where ci > 0 and λi > 0 are constants independent of A.

Proposition 2.4. Let μ be a probability measure on X. Then μ is K-moderate if and only if it isweakly moderate, i.e., there are constants λ > 0 and α > 0 such that∫

exp(λ|u|α)

dμ� c for u ω-psh with maxX

u = 0.

In particular, if μ is moderate, then it is K-moderate.

Proof. Assume that μ is weakly moderate as above. We will show that it is K-moderate. It isenough to obtain the estimate in Definition 2.3 for every compact set A such that capBTK(A) issmall enough. If capBTK(A) = 0, by (1), A is pluripolar, i.e., A ⊂ {u = −∞} for some ω-pshfunction u; hence μ(A) = 0 since μ is weakly moderate. So we can assume that capBTK(A) isstrictly positive. We deduce that T (A) > 0.

Let u be an ω-psh function with maxX u = 0 such that

u � logT (A) + 1 on A.

Using that T (A) is small, we obtain that

μ(A) � μ{u� logT (A) + 1

}� μ

{λ|u|α �

(− logT (A))α/2}

.

Since μ is weakly moderate, if c is the constant as in the lemma, the last quantity is boundedabove by

c exp(−(− logT (A)

)α/2),

which is, by the second inequality in (1), dominated by

c′ exp(− capBTK(A)−α′)

for some constants c′ > 0 and α′ > 0. So μ is K-moderate.

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Assume now that μ is K-moderate as in Definition 2.3. We will show that it is weakly mod-erate. Let u be an ω-psh function such that maxX u = 0. It is enough to show for any M � 0that

μ{u < −M}� c′′ exp(−Mα′′)

for some constants c′′ > 0 and α′′ > 0 independent of M and of u. Let A be an arbitrary compactsubset of the open set {u < −M}. We want to bound μ(A).

We have T (A) � e−M by definition of T . We can assume that capBTK(A) and T (A) aresmall enough. Since μ is K-moderate, we obtain for some constant α′′ > 0

logμ(A)� const − capBTK(A)−α �−(− logT (A))α′′

� −Mα′′,

where the second inequality follows from the first estimate in (1). Hence, μ is weakly moderate.The lemma follows. �Example 2.5. Let μ be a probability measure on P

1 smooth except at 0 and such that

μ = |z|−2 exp(−(− log |z|)1/2)

(i dz ∧ dz) near 0.

In the polar coordinates r := |z| and θ := arg(z) ∈ (−π,π], we have

μ = 2r−1 exp(−(− log r)1/2)dr ∧ dθ.

If u is equal to log |z| near 0, we see that exp(−λu) is not μ-integrable for any λ > 0. So μ is notmoderate. We now check that μ is, however, weakly moderate and hence it is K-moderate.

Since μ is smooth outside the point 0, we can reduce the problem to a local estimate near 0.Since μ is invariant by rotations about 0, using the following lemma, we see that μ satisfies anestimate as in Proposition 2.4 for α = 1/4.

Lemma 2.6. Let D1,D2 be two discs in C of center 0 and of radius r1 and r2 respectively withr1 < r2 � 1. Let F be a family of subharmonic functions on D2 which is bounded in L1(D2).Then there are constants λ > 0 and c > 0 such that

π∫−π

eλ|u(reiθ )|1/4dθ � ce(− log r)1/4

for all 0 < r � r1 and u ∈ F .

Proof. Since u is bounded in L1(D2), by Chern–Levine–Nirenberg’s inequality, the mass ofi∂∂u on a compact set of D2 is bounded, see e.g. [7, Rk. 3.4]. So we can reduce slightly D2 andmultiply each function u with a constant in order to assume that the positive measure ν := i∂∂u

is of mass at most 1.Define

u′(z) := 1

π

∫log |z − w|dν(w).

w∈D2

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This is a potential of ν. So v := u − u′ is a harmonic function with bounded L1-norm on D2. Itfollows that v is bounded on D1. Therefore, since |u|1/4 � |u′|1/4 + |v|1/4, it is enough to provethe estimate in the lemma for the function u′ instead of u.

Now, since the function t �→ et1/4is convex for t real large enough, by Jensen’s inequality

and with u′ defined as above, it is enough to check the inequality in the lemma for the functionu′′ := log |z−w| with |w|� r2 � 1, instead of the function u. Without loss of generality, we canassume that w is real and positive.

Assume that |z| = r � r1. If (z) � 0 we have |z−w| � r � 12 |z− r| and otherwise |z−w| �

|�(z)| � 12 |z − r|. So we always have |z − w|� 1

2 |z − r|. Hence, we have

∣∣u′′(reiθ)∣∣� | log r| + ∣∣log

∣∣eiθ − 1∣∣∣∣ + const � | log r| + | log θ | + const.

It follows that

∣∣u′′(reiθ)∣∣1/4 � | log r|1/4 + | log θ |1/4 + const.

It is now clear that u′′ satisfies the estimate in the lemma for λ = 1 and for c > 0 largeenough. �3. Super-potentials of positive measures

Recall that C denotes the convex compact set of positive closed (1,1)-currents in the coho-mology class {ω}. We can consider for each real number α > 0 the following distance on C :

distα(T ,T ′) := sup

‖Φ‖Cα�1

∣∣⟨T − T ′,Φ⟩∣∣

where Φ is a test smooth (n − 1, n − 1)-form. Observe that the family distα is decreasing in α.The following proposition was obtained in [16] as a consequence of the interpolation theory forBanach spaces.

Proposition 3.1. Let α and β be real numbers such that β � α > 0. Then there is a constantc = c(α,β) > 0 such that

distβ � distα � c(distβ)α/β .

For each real number p > 1, define the following distance on C

distLp

(T ,T ′) := ∥∥u − u′∥∥

Lp ,

where u and u′ are ω-potentials of T and T ′ respectively and the Lp norm is with respect to themeasure ωn. Since we assume that ωn is a probability measure, the family distLp is increasingin p. Note that C has finite diameter with respect to all the above distances. Indeed, for distα thisproperty follows from the fact that the currents in C have a bounded mass, and for distLp we usethe fact that the set of ω-potentials of currents in C is compact in Lp , see e.g. [15].

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Proposition 3.2. Let p � 1 be a real number. Then there are constants c > 0, c′ > 0, and c′′ > 0depending on p such that

c dist2 � distL1 � c′ dist1 and distLp � c′′ max(1,− log distL1)p−1p (distL1)

1p .

Proof. Given T ,T ′ ∈ C , let u and u′ be the ω-potentials of T and T ′ respectively. Let Φ be asmooth (n − 1, n − 1)-form such that ‖Φ‖C 2 � 1. We have∣∣⟨T − T ′,Φ

⟩∣∣ = ∣∣⟨u − u′, ddcΦ⟩∣∣� distL1

(T ,T ′).

So the first inequality in the proposition is clear.Consider the second inequality in the proposition. Let π1 and π2 denote the canonical projec-

tions from X×X onto its factors. Let be the diagonal of X×X. In the proof of [14, Prop. 2.1],an explicit kernel K(x,y) on X × X was obtained, see also Bost, Gillet and Soulé [4]. It is an(n − 1, n − 1)-form smooth outside such that∥∥K(x,y)

∥∥ � − log dist(x, y)dist(x, y)2−2n

and ∥∥∇K(x,y)∥∥ � dist(x, y)1−2n

when (x, y) tends to . Here, ‖∇K(x,y)‖ denotes the sum of the norms of the gradients of thecoefficients of K(x,y) for a fixed atlas on X × X.

This kernel gives a solution v to the equation ddcv = T − T ′ in the sense of currents with

v(x) :=∫

y∈X

K(x, y) ∧ (T (y) − T ′(y)

)

or more formally

v := (π1)∗(K ∧ π∗

2

(T − T ′)).

We refer to [14, Prop. 2.1] for details. The key point here is that if [ ] denotes the current ofintegration on the diagonal , then [ ] − ddcK is a smooth representation of the cohomologyclass of in the Künneth decomposition of Hn,n(X × X,C).

We show that ‖v‖L1 � dist1(T ,T ′). Consider a test smooth (n,n)-form Φ such that‖Φ‖∞ � 1. We have

〈v,Φ〉 =∫

X×X

K(x, y) ∧ Φ(x) ∧ (T (y) − T ′(y)

) = ⟨T − T ′, (π2)∗

(K ∧ π∗

1 (Φ))⟩

.

The estimates on K imply that (π2)∗(K ∧ π∗1 (Φ)) is a form with bounded C 1 norm. We

deduce that |〈v,Φ〉| � dist1(T ,T ′). Since this property holds for every Φ , we obtain that‖v‖L1 � dist1(T ,T ′).

Define m := ∫X

vωn and v := v − m. It follows from the above discussion that |m| �dist1(T ,T ′) and ddcv = T −T ′ = ddc(u−u′). Since the solution of the equation ddcv = T −T ′

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with∫X

vωn = 0 is unique, we obtain that v = u − u′. This, combined with the estimates‖v‖L1 � dist1(T ,T ′) and |m| � dist1(T ,T ′), implies that ‖u − u′‖L1 � dist1(T ,T ′). The sec-ond inequality in the proposition follows.

Recall that the measure ωn is moderate, see e.g. [32, p. 677]. Indeed, the estimates given inthese references show that every measure with bounded density is moderate, see also [12] fora more general result. So Lemma 2.2 applied to ωn implies the last inequality in the proposi-tion. �

Propositions 3.1 and 3.2 show that the distances considered above define the same topologyon C . It is not difficult to see that this topology is induced by the weak topology on currents.The above propositions also imply that the following notion does not depend on the choice of thedistance on C and therefore gives us a large flexibility to prove this Hölder property.

The following elementary lemma gives another characterization of positive measures with aHölder continuous super-potential.

Lemma 3.3. Let μ be a positive measure on X. Then its super-potential U is Hölder continuouswith Hölder exponent 0 < β � 1 with respect to the distance distL1 on C if and only if there is aconstant c > 0 such that∥∥u − u′∥∥

L1(μ)� c max

(∥∥u − u′∥∥L1 ,

∥∥u − u′∥∥β

L1

)for all ω-psh functions u and u′.

Proof. We first prove the necessary condition. Assume that U is Hölder continuous with expo-nent β as above. Observe that it is enough to show that∣∣∣∣

∫ (u − u′)dμ

∣∣∣∣� max(∥∥u − u′∥∥

L1,∥∥u − u′∥∥β

L1

).

Indeed, this inequality applied to u,max(u,u′) and then to u′,max(u,u′) gives the result.Consider first the case where

∫X

uωn = ∫X

u′ωn = 0. Define T := ddcu + ω and T ′ :=ddcu′ + ω. Then, by hypothesis, we have∣∣∣∣

∫ (u − u′)dμ

∣∣∣∣ = ∣∣U (T ) − U(T ′)∣∣� distL1

(T ,T ′)β = ∥∥u − u′∥∥β

L1 .

In the general case, define m := ∫X

uωn and m′ := ∫X

u′ωn. We can apply the first case to v :=u − m and v′ := u′ − m′. In order to obtain the inequality in the lemma, it is enough to use thetriangle inequality and to observe that |m − m′|� ‖u − u′‖L1 .

For the sufficient part, assume the inequality in the lemma. Consider two currents T and T ′in C . Denote by u and u′ their ω-potentials which belong to a fixed compact family of ω-pshfunctions. Therefore, ‖u‖L1 , ‖u′‖L1 and ‖u − u′‖L1 are bounded. The inequality in the lemmaimplies that

∣∣U (T ) − U(T ′)∣∣ =

∣∣∣∣∫ (

u − u′)dμ

∣∣∣∣� ∥∥u − u′∥∥β

L1 .

The lemma follows. �

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The following example gives us a measure with continuous super-potential. However, theassociated Monge–Ampère equation has an unbounded solution.

Example 3.4. Let ω be the Fubini–Study form on P2. Consider a positive measure μ =

(ddcϕ + ω)2 where ϕ is an ω-psh function smooth outside a point a. Consider the convexincreasing function χ(t) := − log(log(−t)) on t < −1. Assume that for some local complexcoordinates z vanishing at a, the function ϕ − χ(log‖z‖) is a potential of −ω. So we haveμ = (ddcχ(log‖z‖))2 near a = 0. We show that the super-potential of μ is continuous.

Fix a smooth function ρ with compact support in a small ball B of center 0 such that ρ = 1in a neighbourhood of 0. Since μ is smooth outside 0, we only have to check that if u is a p.s.h.function on B then 〈μ,ρu〉 depends continuously on u. The continuity is with respect to the L1

loctopology on u.

For this purpose, using again that μ is smooth outside 0, we only have to check that if F isa compact family of p.s.h. functions on B, then for any ε > 0, there is a neighbourhood U of 0such that ∫

U

|u|dμ =∫U

|u|(ddcχ(log‖z‖))2 � ε.

Since μ is invariant under the action of the unitary group U(2), we can replace u by the naturalaverage of u◦ τ with τ ∈ U(2). In other words, we can assume that u is radial. Recall that a radialfunction is p.s.h. if and only if it is a convex increasing function on log‖z‖. Now, since u belongsto a compact family of p.s.h. functions, there is a constant c > 0 such that |u| � c(1 − log‖z‖).Therefore, we only have to check that

−∫B

log‖z‖(ddcχ(log‖z‖))2

is finite.Define r := ‖z‖. The form involved in the last integral is smooth outside 0. Its coefficient is

bounded by a constant times

− log r

r4

[χ ′(log r) + χ ′′(log r)

]2 � − 1

r4 log r(log(− log r))2

which is integrable in a neighbourhood of 0. The result follows. Note that since the consideredform is invariant under U(2) in the last computation, for simplicity, we can assume that z2 = 0.

4. Measures with a Hölder super-potential

In this section we study various sufficient conditions for a positive measure μ to be withHölder continuous super-potential.

Proposition 4.1. Let ϕ be a Hölder continuous ω-psh function on X with Hölder exponent 0 <

α � 1. Then the super-potential of the measure μ := (ddcϕ + ω)n, defined as in Definition 1.2,is Hölder continuous with respect to the distance distL1 on C and with Hölder exponent αn

n ·

(2+α)

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Proof. Define μk := (ddcϕ + ω)k ∧ ωn−k . We prove by induction on k that the super-potential

of μk is Hölder continuous with Hölder exponent μk := αk

(2+α)k· The property is clearly true for

k = 0. Assuming it true for k − 1 with some 1 � k � n, we show that the property also holdsfor k. To do this it is sufficient to check the criterion given in Lemma 3.3. Let u and u′ be twoω-p.s.h. functions. We can assume u � u′ since we can always reduce the problem to the casewith u,max(u,u′) and the case with u′,max(u,u′). Subtracting from u and u′ a constant allowsto assume that

∫X

uωn = 0.Define also m′ := ∫

Xu′ωn and u′ := u′ − m′. So u and u′ belong to a fixed compact family

of ω-psh functions. We deduce that ‖u‖L1 and ‖u′‖L1 are bounded. By [7, Ch. III (3.11)], theintegrals 〈μ,u〉 and 〈μ, u′〉 are also bounded. Therefore, by Lemma 3.3, we only have to considerthe case where |m′| is bounded by a fixed constant large enough and to prove the inequality∫ (

u − u′)dμk �∥∥u − u′∥∥βk

L1 .

Indeed, ‖u − u′‖L1 is bounded by a fixed constant.Since ϕ is Hölder continuous, using a standard convolution and a partition of unity, we can

write ϕ = ϕε + (ϕ − ϕε) with ‖ϕε‖C 2 � ε−2 and |ϕ − ϕε | � εα for some α > 0. We have forT := ddcu + ω and T ′ := ddcu′ + ω

∣∣⟨μk,u − u′⟩∣∣� ∣∣⟨μk−1, u − u′⟩∣∣ + ∣∣⟨ddcϕ ∧ (ddcϕ + ω

)k−1 ∧ ωn−k, u − u′⟩∣∣�

∣∣⟨μk−1, u − u′⟩∣∣ + ∣∣⟨ddcϕε ∧ (ddcϕ + ω

)k−1 ∧ ωn−k, u − u′⟩∣∣+ ∣∣⟨(ϕ − ϕε) ∧ (

ddcϕ + ω)k−1 ∧ ωn−k, T − T ′⟩∣∣.

Since u − u′ � 0 and ±ddcϕε are bounded by a constant times ε−2ω, the second term inthe last sum is bounded by a constant times ε−2|〈μk−1, u − u′〉|. Applying the Chern–Levine–Nirenberg inequality [7, Ch. III (3.3)] to the last term in the above sum, we see that this term isbounded by a constant times εα . This together with the induction hypothesis yields

∣∣⟨μk,u − u′⟩∣∣� ε−2∣∣⟨μk−1, u − u′⟩∣∣ + εα � ε−2

∥∥u − u′∥∥βk−1

L1 + εα.

Choosing ε equal to a fixed constant small enough times ‖u − u′‖βk−12+α

L1 = ‖u − u′‖βkα

L1 , the proofis thereby complete.

Note that we can show in the same way that a wedge-product of positive closed currents (ofarbitrary bidegree) with Hölder continuous super-potentials is also of Hölder continuous super-potential. �

Let ϕ be a fixed smooth function on Rn with compact support in the ball of center 0 and of

radius 3/2 which is equal to 1 on a neighbourhood of the unit ball. Define

ϕ0 := ϕ, ϕ1(x) := ϕ

(x

2

)− ϕ(x) and ϕj (x) = ϕ1

(2−j+1x

)for x ∈R

n, j ∈N.

Denote by u and u the Fourier and the inverse Fourier transforms of u respectively.

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Recall from [30, p. 15] that the Besov space Bs∞,∞(Rn) with s ∈ R can be defined by

Bs∞,∞(R

n) :=

{u tempered distribution on R

n: supj∈N

2js∥∥(ϕj u)ˇ∥∥

L∞ < ∞},

where a tempered distribution on Rn is a continuous linear form on the Schwartz space on R

n.The Hölder space C ε(Rn) for 0 < ε < 1 coincides with the Besov space Bε∞,∞(Rn), see [30,

p. 4]. The above definition of Besov spaces does not depend on the choice of ϕ and it extends ina natural way to the case of compact manifolds using a partition of unity.

The following result, together with Theorem 1.3, implies Corollary 1.4.

Proposition 4.2. Let μ be a positive measure on X. Assume that locally we can write μ =ddcU + ∂V + ∂W with Hölder continuous forms U,V,W of bidegrees (n − 1, n − 1), (n −1, n) and (n,n − 1) respectively. Then μ is of Hölder continuous super-potential. Moreover, thehypothesis on μ is satisfied when μ belongs to the Sobolev space W 2n/p−2+ε,p(X) or to theBesov space Bε−2∞,∞(X) for some ε > 0 and p > 1.

Proof. Consider a coordinate ball B in X and a smooth positive function χ with compact supportin B. We can assume that μ = ddcU + ∂V + ∂W as above on B. Using a partition of unity, it isenough to show that μ′ := χ(ddcU + ∂V + ∂W) is of Hölder continuous super-potential.

For 0 < ε � 1, using the standard convolution, we can write U = Uε + (U − Uε) with‖Uε‖C 2 � ε−2 and ‖U − Uε‖∞ � εα for some constant α > 0. We obtain in the same waythe regularizing forms Vε and Wε of V and W respectively. Using these estimates, we have

∣∣⟨μ′, u − u′⟩∣∣� ∣∣⟨ddcU,χ(u − u′)⟩∣∣ + ∣∣⟨∂V,χ

(u − u′)⟩∣∣ + ∣∣⟨∂W,χ

(u − u′)⟩∣∣

�∣∣⟨ddcUε,χ

(u − u′)⟩∣∣ + ∣∣⟨U − Uε,ddc

(χ

(u − u′))⟩∣∣

+ ∣∣⟨∂Vε,χ(u − u′)⟩∣∣ + ∣∣⟨V − Vε, ∂

(χ

(u − u′))⟩∣∣

+ ∣∣⟨∂Wε,χ(u − u′)⟩∣∣ + ∣∣⟨W − Wε, ∂

(χ

(u − u′))⟩∣∣.

Recall that ‖du‖L1 and ‖du′‖L1 are bounded by a constant. These properties are consequencesof classical properties of p.s.h. functions. We can also obtain them using the estimates on K and∇K given in Proposition 3.2. Consequently, ddc(χ(u − u′)), ∂(χ(u − u′)) and ∂(χ(u − u′))have bounded mass. This discussion, combined with the above estimates on |〈μ′, u − u′〉| andthe above mentioned properties of Uε , Vε , Wε , implies that∣∣⟨μ′, u − u′⟩∣∣� ε−2

∥∥u − u′∥∥L1 + εα.

To obtain the first assertion in the proposition it is enough to take ε := ‖u − u′‖1

2+α

L1 .For the second assertion, locally on a small ball B in X with holomorphic coordinates z, if u is

a solution of the Laplacian equation u = μ and if U is a suitable constant times u(ddc‖z‖2)n−1,then ddcU = μ. When μ belongs to the Sobolev space W 2n/p−2+ε,p(X), the function u is inW 2n/p+ε,p(B), see [1, p. 254] for the definition of Ws,p with s ∈ R. By Sobolev’s embeddingtheorem, u is Hölder continuous, see e.g. [1, p. 85]. When μ is in the Besov space Bε−2∞,∞(X),the solution u is in Bε∞,∞(X), see [30, p. 255]. The last space coincides with the Hölder spaceC ε(X). The result follows. �

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The following result and Theorem 1.3 imply the main result of Hiep in [23].

Proposition 4.3. Let μ be a positive measure on X. Assume that there are constants c > 0 andα > 0 such that if B is a ball of radius r in X we have μ(B) � cr2n−2+α . Then the super-potentialof μ is Hölder continuous.

Proof. Let T ,T ′, u,u′, the super-potential U , the kernel K , the function v and the constant m

be given in Proposition 3.2 above. We have

U (T ) − U(T ′) = ⟨

μ,u − u′⟩ = 〈μ,v − m〉= ⟨

μ, (π1)∗(K ∧ π∗

2

(T − T ′))⟩ − m

= ⟨T − T ′, (π2)∗

(K ∧ π∗

1 (μ))⟩ − m.

Since we already have good estimates on m, it is enough to check that the form Φ := (π2)∗(K ∧π∗

1 (μ)) is Hölder continuous.Let B be a coordinate ball in X that we identify with the unit ball in C

n. We will show that Φ

is Hölder continuous near the origin 0 ∈ Cn. Let χ be a smooth function with compact support

in B×B and equal to 1 near the origin. We have

Φ = (π2)∗(χK ∧ π∗

1 (μ)) + (π2)∗

((1 − χ)K ∧ π∗

1 (μ)).

Since the form (1−χ)K is smooth near X ×{0}, the last expression in the above identity definesa smooth form near 0. It remains to show that Ψ := (π2)∗(χK ∧ π∗

1 (μ)) is Hölder continuousnear 0.

Observe that the coefficients of Ψ have the form

Θ(x) :=∫

y∈BH(x,y) dμ(y)

where H is a coefficient of χK . We use now the estimates on K and ∇K given in Proposition 3.2.Consider two points x and x′ in B near 0 and denote by D the ball of center x and of radiusρ := 2‖x − x′‖1/(2n). We have

Θ(x) − Θ(x′) =

∫D

H(x, y) dμ(y) −∫D

H(x′, y

)dμ(y) +

∫B\D

(H(x,y) − H

(x′, y

))dμ(y).

The estimate on K and the hypothesis on μ imply that the first two terms are bounded by aconstant times | logρ|ρα . The estimate on ∇K implies that∣∣H(x,y) − H

(x′, y

)∣∣� ∥∥x − x′∥∥ρ1−2n for y /∈ D.

Therefore, the last integral in the above identity is bounded by a constant times ‖x − x′‖1/(2n).We deduce that Θ is a Hölder continuous function. The proposition follows. �Proposition 4.4. Let μ be a positive measure on X with Hölder continuous super-potential. Thenμ is moderate.

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Proof. Let u be an ω-psh function with∫X

uωn = 0 and set uM := max(u,−M) for M > 1 largeenough. By Lemma 2.2 applied to the measure ωn, there is a constant α > 0 independent of u

and M such that ∣∣∣∣∫X

uMωn

∣∣∣∣ =∣∣∣∣∫X

(uM − u)ωn

∣∣∣∣ � e−αM.

Denote by β a Hölder exponent of U with respect to the distance distL1 on C with 0 < β � 1.Since M is large enough, the estimate above shows that ‖u − uM‖L1 is small. We deduce fromLemma 3.3 and the inequality uM − u > 1 on {u < −M − 1} that

μ{u < −M − 1} �∣∣∣∣∫

(u − uM)dμ

∣∣∣∣� ‖u − uM‖β

L1 � e−αβM.

Thus, μ is moderate. �The following corollary gives us a large family of measures with Hölder continuous super-

potential. It shows that the restriction of such a measure to a Borel set gives also a measure withHölder continuous super-potential. Note that by definition the set of measures with Hölder con-tinuous super-potential is a convex cone. We then deduce from Theorem 1.3 analogous propertiesfor Monge–Ampère measures with Hölder continuous potential which have been obtained in [8].

Corollary 4.5. Let μ be a positive measure with a Hölder continuous super-potential. If f isa positive function in Lp(μ) with p > 1, then f μ is a measure with Hölder continuous super-potential.

Proof. Let q � 1 be the real number such that p−1 + q−1 = 1. Let V be the super-potential off μ. Consider two currents T ,T ′ in C and their ω-potentials u,u′. We have∣∣V (T ) − V

(T ′)∣∣ = ∣∣⟨f μ,u − u′⟩∣∣� ‖f ‖Lp(μ)

∥∥u − u′∥∥Lq(μ)

.

By Proposition 4.4 and Lemma 2.2, the last expression is bounded by a constant times‖u − u′‖1/(2q)

L1(μ). Lemma 3.3 implies the result. �

5. End of the proof of Theorem 1.3

The necessary condition follows from Proposition 4.1. To prove the sufficient part assume thatthe super-potential U of μ is Hölder continuous with Hölder exponent 0 < β � 1 with respect tothe distance distL1 on C . Propositions 4.4 and 2.4 imply that for every p > 0 there is a constantc > 0 such that

μ(A)� c capBTK(A)p for any compact set A.

In other words, μ satisfies the condition H(∞) in the sense of [18] (see also Definition 2.6in [8]) which is stronger than condition (A) in [24]. The latter work ensures the existence of abounded ω-psh solution u to (ddcu+ω)n = μ with the normalization infX u = 1 (for the reader’sconvenience we use here the notation u instead of ϕ as in Ref. [8] that we use now).

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For δ > 0 small enough, let ρδu denote the regularization of u defined in [8, Sec. 2], see also[6]. It satisfies

‖ρδu − u‖L1 � cδ2 and ddcρδu� −cω

for some constant c � 1. Lemma 3.3 applied to the ω-psh functions c−1ρδu and c−1u impliesthat ∫

X

|ρδu − u|dμ� ‖ρδu − u‖β

L1 � δ2β. (2)

We are now able to apply Proposition 3.3 in [8] which shows that u is Hölder continuous.Now, we follow along the same lines as those given in the proof of Theorem A in [8] in order

to get an explicit Hölder exponent of u in terms of n and β . It is enough to make the followingchanges:

• After (3.1) in [8]: let q := 1, 0 < α1 <2β

n+1 and choose ε > 0, α, α0 such that α1 < α < α0 <

2β −α0(n+ ε). Take f ≡ 1 and set g := 0 on E0 and g := c elsewhere with a constant c � 1such that ‖gμ‖ = ‖μ‖.

• As in [11,25], we solve for continuous ω-psh function v(ddcv + ω

)n = gμ with max(u − v) = max(v − u).

• (3.2) in [8] becomes now (2) which implies∫E0

dμ� c2δ2β−α0 with c2 > 0. The last relation

replaces the inequality∫E0

f ωn � c2δ2−α0

q in [8].

We obtain that u is Hölder continuous with exponent α1 for all 0 < α1 <2β

n+1 . �Recall that by Proposition 4.1, when ϕ is Hölder continuous with Hölder exponent 0 < α � 1,

the super-potential of μ = (ddcϕ + ω)n is Hölder continuous of order αn

(2+α)nwith respect to

distL1 on C . Conversely, we see in the above proof that if the super-potential of μ is Höldercontinuous with exponent 0 < β � 1, then the solution ϕ is Hölder continuous with exponent

2βn+1 . It would be interesting to investigate the sharp exponent estimates. Note that for some steps

in the proof, the obtained constant 2βn+1 is optimal, see [8,10] for details.

Acknowledgments

The authors thank Nicolas Lerner and Nessim Sibony for their help during the preparation ofthis work. They also thank the referee for his remarks and suggestions.

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