arXiv:0704.0359v1  [math.CV]  3 Apr 2007
Some properties of the complex Monge-Ampère operator in
Cegrell’s classes and applications
NGUYEN VAN KHUE and PHAM HOANG HIEP
Abstract. In this article we will first prove a result about convergence in capacity. Using
the achieved result we will obtain a general decompositon theorem for complex Monge-
Ampère measues which will be used to prove a comparison principle for the complex
Monge-Ampère operator.
2000 Mathematics Subject Classification: Primary 32W20, Secondary 32U15.
Key words and phrases: complex Monge-Ampère operator, plurisubharmonic function.
This work was supported by the National Research Program for Natural Sciences, Vietnam.
1. Introduction
Let Ω be a bounded hyperconvex domain in Cn. By PSH(Ω) we denote the set of plurisub-
harmonic (psh) functions on Ω. In [BT 1,2] the authors established and used the compari-
son principle to study the Dirichlet problem in PSH∩L∞loc(Ω). Recently, Cegrell introduced
a general class E of psh functions on which the complex Monge-Ampère operator (ddc.)n
can be defined. He obtained many important results of pluripotential theory in the class E .
For example, the ones on the comparison principle and solvability of the Dirichlet problem
(see [Ce 1-3]).
The main result of our paper are Theorem 4.1 and some Xing type comparision principles.
Theorem 4.1 is generalize Lemma 5.4 in [Ce1], Lemma 7.2 in [Åh] and Lemma 3.4 in [Ce3].
For definitions of Cegrell’s classes see Section 2. After giving some preliminaries, we start in
Proposition 3.1 with a comparison principle, which is analogous to a comparison principle
due to Xing (Lemma 1 in [Xi1]). It should be observed that our proof is quite different
from Xing’s proof, and the inequality we obtain is slightly stronger than Xing’s inequality,
even in the case of bounded psh functions. Using Proposition 3.1, we give in Theorem
3.5 a sufficient condition for Cn-capacity convergence of a sequence of psh functions in
the class F . This result should be compared to Theorem 3 of [Xi1] where the situation
of bounded psh functions was studied. Applying Theorem 3.5 we give generalizations of
recent results in [Cz] and [CLP] about convergences of multipole Green functions and a
criterion for pluripolarity, respectively. Section 4 focuses on Theorem 4.1 and Theorem
4.9. By applying Theorem 4.1 we give some results on class Cegrell’s classes. We prove in
Proposition 4.4 a local estimate for the Monge-Ampère measure in terms of the Beford-
Taylor relative capacity. As an application, we give in Theorem 4.5 a decomposition result
for Monge-Ampère measure, which is similar in spirit to Theorem 6.3 in [Ce1]. From
Proposition 3.1 and Theorem 4.1 we obtain easily a Xing type comparison principle for
functions in classes F and E .
Acknowledgment. We are grateful to Professor Urban Cegrell for useful discussions that
helped to improve the paper. We are grateful to Per Åhag for fruitful comments. This
work is supported by the National Research Program for Natural Sciences, Vietnam.
http://arxiv.org/abs/0704.0359v1
2. Preliminaries
First we recall some elements of pluripotential theory that will be used throughout the
paper. All this can be found in [BT2], [Ce1], [Ce2], [Le].
2.1. We will always denote by Ω a bounded hyperconvex domain in Cn unless other wise
stated. The Cn-capacity in the sense of Bedford and Taylor on Ω is the set function given
Cn(E) = Cn(E,Ω) = sup{
(ddcu)n : u ∈ PSH(Ω), −1 ≤ u ≤ 0}
for every Borel set E in Ω. It is proved in [BT2] that
Cn(E) =
(ddch∗E,Ω)
where h∗E,Ω is the upper regularization of the relative extremal function hE,Ω for E (relative
to Ω) i.e.,
hE,Ω(z) = sup{u(z) : u ∈ PSH
−(Ω), u ≤ −1 on E}.
The following concepts are taken from [Xi1] and [Xi2]
∗A sequence of functions uj on Ω is said to converge to a function u in Cn-capacity on a
set E ⊂ Ω if for every δ > 0 we have Cn({z ∈ E : |uj(z) − u(z)| > δ}) → 0 as j → ∞.
∗A family of positive measures {µα} on Ω is called uniformly absolutely continuous with
respect to Cn-capacity in a set E ⊂ Ω if for every ǫ > 0 there exists δ > 0 such that for
each Borel subset F ⊂ E with Cn(F)< δ the inequality µα(F)< ǫ holds for all α. We write
µα ≪ Cn in E uniformly for α.
2.2. The following classes of psh functions were introduced by Cegrell in [Ce1] and [Ce2]
E0 = E0(Ω) = {ϕ ∈ PSH
−(Ω) ∩ L∞(Ω) : lim
ϕ(z) = 0,
(ddcϕ)n < +∞},
F = F(Ω) = {ϕ ∈ PSH−(Ω) : ∃ E0(Ω) ∋ ϕj ց ϕ, sup
(ddcϕj)
n < +∞},
E = E(Ω) = {ϕ ∈ PSH−(Ω) : ∃ ϕK ∈ F(Ω) such that ϕK = ϕ on K, ∀K ⊂⊂ Ω},
Ea = Ea(Ω) = {u ∈ E(Ω) : (ddcu)n(E) = 0 ∀ E is pluripolar in Ω}.
For each u ∈ F(Ω), we set
e0(u) =
(ddcu)n.
2.3. Let A = {(wj , νj)}j=1,...,p be a finite subset of Ω × R
+. According to Lelong (see
[Le]), the pluricomplex Green function with poles in A is defined by
g(A)(z) = sup{u(z) : u ∈ LA}
where
LA = {u ∈ PSH
−(Ω) : u(z) − νj log |z − wj | ≤ O(1) as z → wj , j = 1, ..., p}
ν(A) =
νnj , Â = {wj}j=1,...,p.
2.4. We write lim
[u(z) − v(z)] ≥ a if for every ǫ > 0 there exists a compact set K in Ω
such that
u(z) − v(z) ≥ a− ǫ for z ∈ (Ω\K) ∩ {u > −∞}
v(z) = −∞ for z ∈ (Ω\K) ∩ {u = −∞}.
2.5. Xing’s comparison principle (see Lemma 1 in [Xi1]). Let Ω be a bounded open subset
in Cn and u, v ∈ PSH∩L∞(Ω) satisfy lim
[u(z)− v(z)] ≥ 0. Then for any constant r ≥ 1
and all wj ∈ PSH(Ω) with 0 ≤ wj ≤ 1, j = 1, 2, ..., n we have
(n!)2
{u<v}
(v − u)nddcw1 ∧ ... ∧ dd
{u<v}
(r − w1)(dd
v)n ≤
{u<v}
(r − w1)(dd
3. Some convergence theorems
In order to study the convergence of a sequence of psh functions in Cn-capacity, we start
with the following.
3.1. Proposition. a) Let u, v ∈ Fsuch that u ≤ v on Ω. Then for 1 ≤ k ≤ n
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cv)k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
for all wj ∈ PSH(Ω), 0 ≤ wj ≤ 1, j = 1, ..., k, wk+1, ..., wn ∈ F and all r ≥ 1.
b) Let u, v ∈ E such that u ≤ v on Ω and u = v on Ω\K for some K ⊂⊂ Ω. Then for
1 ≤ k ≤ n
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cv)k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
for all wj ∈ PSH(Ω), 0 ≤ wj ≤ 1, j = 1, ..., k, wk+1, ..., wn ∈ E and all r ≥ 1.
We proceed through some lemmas.
3.2. Lemma. Let u, v ∈ PSH ∩ L∞(Ω) such that u ≤ v on Ω and lim
[u(z) − v(z)] = 0.
(v − u)kddcw ∧ T ≤ k
(1 − w)(v − u)k−1ddcu ∧ T
for all w ∈ PSH(Ω), 0 ≤ w ≤ 1 and all positive closed currents T .
Proof. First, assume u, v ∈ PSH∩L∞(Ω), u ≤ v on Ω and u = v on Ω\K, K ⊂⊂ Ω. Then,
using the Stokes formula we obtain
(v − u)kddcw ∧ T =
(v − u)kddc(w − 1) ∧ T
(w − 1)ddc(v − u)k ∧ T
= −k(k − 1)
(1 − w)d(v − u) ∧ dc(v − u) ∧ T
(1 − w)(v − u)k−1ddc(u− v) ∧ T
(1 − w)(v − u)k−1ddc(u− v) ∧ T
(1 − w)(v − u)k−1ddcu ∧ T.
General case, for each ǫ > 0 we set vǫ = max(u, v − ǫ). Then vǫ ր v on Ω, vǫ ≥ u on Ω
and vǫ = u on Ω\K for some K ⊂⊂ Ω. Hence
(vǫ − u)
kddcw ∧ T ≤ k
(1 − w)(vǫ − u)
k−1ddcu ∧ T.
Since 0 ≤ vǫ − u ր v − u as ǫ ց 0, letting ǫ ց 0 we get
(v − u)kddcw ∧ T ≤ k
(1 − w)(v − u)k−1ddcu ∧ T.
3.3. Lemma. Let u, v ∈ PSH ∩ L∞(Ω) such that u ≤ v on Ω and lim
[u(z) − v(z)] = 0.
Then for 1 ≤ k ≤ n
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cv)k ∧ T
(r − w1)(dd
cu)k ∧ T.
for all w1, ..., wk ∈ PSH(Ω), 0 ≤ wj ≤ 1 ∀ j = 1, ..., k, wk+1, ..., wn ∈ E and all r ≥ 1.
Proof. To simplify the notation we set
T = ddcwk+1 ∧ ... ∧ dd
First, assume that u, v ∈ PSH ∩ L∞(Ω), u ≤ v on Ω, and u = v on Ω\K, K ⊂⊂ Ω. Using
Lemma 3.2 we get
(v − u)kddcw1 ∧ ... ∧ dd
cwn ≤ k
(v − u)k−1ddcw1 ∧ ... ∧ dd
cwk−1 ∧ dd
cu ∧ T
≤ ...
(v − u)ddcw1 ∧ (dd
cu)k−1 ∧ T
(v − u)ddcw1 ∧ [
(ddcu)i ∧ (ddcv)k−i−1] ∧ T
(w1 − r)dd
c(v − u) ∧ [
(ddcu)i ∧ (ddcv)k−i−1] ∧ T
(r − w1)dd
c(u− v) ∧ [
(ddcu)i ∧ (ddcv)k−i−1] ∧ T
(r − w1)[(dd
cu)k − (ddcv)k] ∧ T.
General case, for each ǫ > 0 we put vǫ = max(u, v − ǫ). Then vǫ ր v on Ω, vǫ ≥ u on Ω
and vǫ = u on Ω\K for some K ⊂⊂ Ω. Hence
(vǫ − u)
kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
k ∧ T
(r − w1)(dd
u)k ∧ T.
Observe that 0 ≤ vǫ − u ր v− u and (dd
k ∧ T → (ddcv)k ∧ T weakly as ǫ ց 0, r−w1
is lower semicontinuous, by letting ǫ ց 0 we have
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cv)k ∧ T
(r − w1)(dd
cu)k ∧ T.
The proof is finished.
Proof of Proposition 3.1. a) Let E0 ∋ uj ց u and E0 ∋ vj ց v as in the definition of F .
Replace vj by max(uj , vj) we may assume that uj ≤ vj for j ≥ 1. By Lemma 3.3 we have
(vj − ut)
kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
k ∧ ddcwk+1 ∧ ... ∧ dd
for t ≥ j ≥ 1. By Proposition 5.1 in [Ce2] letting t → ∞ in the above inequality we have
(vj − u)
w1 ∧ ... ∧ dd
(r − w1)(dd
k ∧ T
(r − w1)(dd
cu)k ∧ T
for j ≥ 1. Next letting j → ∞ again by Proposition 5.1 in [Ce2] we get the desired
conclusion.
b) Let G,W be open sets such that K ⊂⊂ G ⊂⊂ W ⊂⊂ Ω. According to the remark
following Definition 4.6 in [Ce2] we can choose a function ṽ ∈ F such that ṽ ≥ v and ṽ = v
on W . Set
u on G
ṽ on Ω\G
Since u = v = ṽ on W\K we have ũ ∈ PSH−(Ω). It is easy to see that ũ ∈ F , ũ ≤ ṽ and
ũ = u on W . By a) we have
(ṽ − ũ)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cṽ)k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
ũ)k ∧ ddcwk+1 ∧ ... ∧ dd
Since ũ = ṽ on Ω\G we have
(ṽ − ũ)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cṽ)k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
cũ)k ∧ ddcwk+1 ∧ ... ∧ dd
Since ũ = u, ṽ = v on W and u = v on Ω\K we obtain
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cv)k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
3.4. Proposition. Let u, v ∈ F and u ≤ v on Ω. Then
(v − u)nddcw1 ∧ ... ∧ dd
(−w1)[(dd
u)n − (ddcv)n]
for all wj ∈ PSH(Ω), −1 ≤ wj ≤ 0, j = 1, ..., n.
Proof. The proposition follows from Proposition 3.1 with k = n, r = 1 and wj are replaced
by wj + 1.
3.5. Theorem. Let u, uj ∈ F and uj ≤ u for j ≥ 1. Assume that sup
(ddcuj)
n < +∞
and ||(ddcuj)
n − (ddcu)n||E → 0 as j → ∞ for all E ⊂⊂ Ω. Then uj → u in Cn-capacity
on every E ⊂⊂ Ω as j → ∞.
Proof. Let Ω′ ⊂⊂ Ω and δ > 0. Put
Aj = {z ∈ Ω′ : |uj − u| ≥ δ} = {z ∈ Ω′ : u− uj ≥ δ}.
We prove that Cn(Aj) → 0 as j → ∞. Given ǫ > 0. By quasicontinuity of u and uj , there
is an open set G in Ω such that Cn(G) < ǫ, and uj |Ω\G, u|Ω\G are continuous. We have
Aj = Bj ∪ {z ∈ G : u− uj ≥ δ}.
where Bj = {z ∈ Ω′\G : u− uj ≥ δ} are compact sets in Ω and
Cn(Aj) ≤ lim
Cn(Bj) + ǫ
We claim that lim
Cn(Bj) = 0. By Proposition 3.4 we have
Cn(Bj) =
(ddch∗Bj )
(u− uj)
n(ddch∗Bj )
(−h∗Bj )[(dd
n − (ddcu)n]
{||(ddcuj)
n − (ddcu)n||K +
(−hΩ′)[(dd
n + (ddcu)n]}
{||(ddcuj)
n − (ddcu)n||K + sup
|hΩ′ |[sup
(ddcuj)
(ddcu)n]}.
As lim
hΩ′(z) = 0 there exists K ⊂⊂ Ω such that
|hΩ′ |[sup
(ddcuj)
(ddcu)n] < ǫ.
By the hypothesis
||(ddcuj)
n − (ddcu)n||K < ǫ for j > j0.
Cn(Bj) < 2ǫ for j > j0.
This proves the claim and hence the theorem.
As an application of Theorem 3.5 we have the following
3.6. Proposition. Let g(Aj) be multipolar Green functions on Ω such that
Âj = {w
1, ..., w
} → ∂Ω and sup
ν(Aj) = sup
)n < +∞
Then g(Aj) → 0 as j → ∞ in Cn-capacity.
Proof. By the hypothesis we have
(ddcg(Aj))
n(Ω) = sup
ν(Aj) < +∞
||(ddcg(Aj))
n||K → 0 as j → ∞ for all K ⊂⊂ Ω.
Theorem 3.5 implies that g(Aj) → 0 as j → ∞ in Cn-capacity.
This section ends up with a criterion for pluripolarity
3.7. Theorem. Let uj ∈ F such that sup
(ddcuj)
n < +∞.
Then there is a constant A > 0 such that
i)( lim
∗ ∈ F .
ii)Cn({z ∈ Ω : ( lim
∗(z) < −t}) ≤ A
iii){z ∈ Ω : lim
uj(z) = −∞} is pluripolar.
Proof. i) For each j ≥ 1 put vj = sup{uj, uj+1, ...}. By [Ce2] v
j ∈ F and
(ddcv∗j )
n ≤ sup
(ddcuj)
n < +∞.
By [Ce2] we have v∗j ց v ∈ F .
ii) By Proposition 3.1 in [CKZ] we have
Cn{z ∈ Ω : ( lim
∗(z) < −t} = Cn{z ∈ Ω : v(z) < −t} ≤
2ne0(v)
where A = 2ne0(v).
iii) According to [BT2] we have
Cn{z ∈ Ω : lim
uj(z) = −∞} = Cn{z ∈ Ω : v(z) = −∞} = 0.
Remark. Theorem 3.7 in the case where uj are multipole Green functions was proved by
D.Coman, N.Levenberg and A.Poletsky in Theorem 4.1 of [CLP].
4. Some properties of the Cegrell’s classes and applications
In this section, first we prove the following
4.1. Theorem. Let u, u1, ..., un−1 ∈ E , v ∈ PSH
−(Ω) and T = ddcu1 ∧ ... ∧ dd
cun−1.
ddc max(u, v) ∧ T |{u>v} = dd
cu ∧ T |{u>v}.
We need the following well-known fact.
4.2. Lemma. Let µ be a measure on Ω and f : Ω → R a measurable function on Ω. The
following are equivalent
i)µ(E) = 0 for all Borell sets E ⊂ {f 6= 0}.
fdµ = 0 for every measurable set E in Ω.
Proof. i)⇒ii) follows from:
fdµ =
E\{f=0}
fdµ +
E∩{f=0}
fdµ = 0
ii)⇒i). It suffices to show that µ = 0 on every Xδ = {f > δ > 0}. By the Hahn
decomposition theorem, there exist measurable subsets X+
and X−
of Xδ such that Xδ =
= ∅ and µ ≥ 0 on X+
, µ ≤ 0 on X−
. We have
δµ(X+
fdµ = 0
δµ(X−
fdµ = 0
Hence, µ(X+
) = µ(X−
) = 0. Therefore, we have µ = 0 on Xδ.
Proof of Theorem 4.1.
a) First we prove the proposition for v ≡ a < 0. According to the remark following
Definition 4.6 in [Ce2], without loss of generality we may assume that u, u1, ..., un−1 ∈ F .
Using Theorem 2.1 in [Ce2] we can find
E0 ∩ C(Ω̄) ∋ u
j ց u, E0 ∩ C(Ω̄) ∋ u
ց uk, k = 1, ..., n− 1.
Since {uj > a} is open we have
ddc max(uj, a) ∧ Tj |{uj>a} = dd
cuj ∧ Tj |{uj>a}.
Thus from the inclusion {u > a} ⊂ {uj > a} we obtain
ddc max(uj , a) ∧ Tj |{u>a} = dd
cuj ∧ Tj |{u>a}.
where Tj = dd
1 ∧ ... ∧ dd
n−1. By Corollary 5.2 in [Ce2], it follows that
max(u− a, 0)ddc max(uj , a) ∧ Tj → max(u− a, 0)dd
c max(u, a) ∧ T.
max(u− a, 0)ddcuj ∧ Tj → max(u− a, 0)dd
cu ∧ T.
Hence
max(u− a, 0)[ddc max(u, a) ∧ T − ddcu ∧ T ] = 0.
Using Lemma 4.2 we have
ddc max(u, a) ∧ T = ddcu ∧ T on {u > a}.
b) Assume that v ∈ PSH−(Ω). Since {u > v} =
{u > a > v}, it suffices to show
ddc max(u, v) ∧ T = ddcu ∧ T on {u > a > v}
for all a ∈ Q−. Since max(u, v) ∈ E , by a) we have
ddc max(u, v) ∧ T |{max(u,v)>a} = dd
c max(max(u, v), a) ∧ T |{max(u,v)>a}
= ddc max(u, v, a) ∧ T |{max(u,v)>a}.
(2) ddcu ∧ T |{u>a} = dd
c max(u, a) ∧ T |{u>a}.
Since max(u, v, a) = max(u, a) on set open {a > v} , we have
(3) ddc max(u, v, a) ∧ T |{a>v} = dd
c max(u, a) ∧ T |{a>v}.
Since {u > a > v} ⊂ {u > a}, {a > v}, {max(u, v) > a} and (1), (2), (3) we have
ddc max(u, v) ∧ T |{u>a>v} = dd
cu ∧ T |{u>a>v}.
The next result is an analogue of an inequality due to Demaily in [De2]
4.3. Proposition. a) u, v ∈ E such that (ddcu)n({u = v = −∞}) = 0. Then
(ddc max(u, v))n ≥ 1{u≥v}(dd
cu)n + 1{u<v}(dd
where 1E denotes the characteristic function of E.
b) Let µ be a positive measure which vanishes on all pluripolar subsets of Ω. Suppose
u, v ∈ E such that (ddcu)n ≥ µ, (ddcv)n ≥ µ. Then (ddc max(u, v))n ≥ µ.
Proof. a) For each ǫ > 0 put Aǫ = {u = v − ǫ}\{u = v = −∞}. Since Aǫ ∩ Aδ = ∅ for
ǫ 6= δ there exists ǫj ց 0 such that (dd
cu)n(Aǫj ) = 0 for j ≥ 1. On the other hand, since
(ddcu)n({u = v = −∞}) = 0 we have (ddcu)n({u = v− ǫj}) = 0 for j ≥ 1. Since Theorem
4.1 it follows that
(ddc max(u, v − ǫj))
n ≥ (ddc max(u, v − ǫj))
n|{u>v−ǫj} + (dd
c max(u, v − ǫj))
n|{u<v−ǫj}
= (ddcu)n|{u≥v−ǫj} + (dd
cv)n|{u<v−ǫj}
= 1{u≥v−ǫj}(dd
cu)n + 1{u<v−ǫj}(dd
≥ 1{u≥v}(dd
cu)n + 1{u<v−ǫj}(dd
cv)n.
Letting j → ∞ and by Remark under Theorem 5.15 in [Ce2] we get
(ddc max(u, v))n ≥ 1{u≥v}(dd
cu)n + 1{u<v}(dd
because max(u, v − ǫj) ր max(u, v) and 1{u<v−ǫj} ր 1{u<v} as j → ∞.
b) Argument as a)
4.4. Proposition. Let u1, ..., uk ∈ PSH(Ω) ∩ L
∞(Ω) and uk+1, ..., un ∈ E . Then
ddcu1 ∧ ... ∧ dd
cun = O((Cn(B))
n ) for all Borel sets B ⊂ Ω′ ⊂⊂ Ω.
B(a,r)
ddcu1 ∧ ... ∧ dd
cun = o((Cn(B(a, r)))
n ) as r → 0 for all a ∈ Ω.
where B(a, r) = {z ⊂ Cn : |z − a| < r}
Proof. We may assume that 0 ≤ uj ≤ 1 for j = 1, ..., k. On the other hand, by the remark
following Defintion 4.6 in [Ce2] we again may assume that uk+1, ..., un ∈ F .
i) For each open set B ⊂⊂ Ω, applying Proposition 3.1 we get
ddcu1 ∧ ... ∧ dd
cun =
(−h∗B)
kddcu1 ∧ ... ∧ dd
(−h∗B)
kddcu1 ∧ ... ∧ dd
(1 − u1)(dd
ch∗B)
k ∧ ddcuk+1 ∧ ... ∧ dd
(ddch∗B)
k ∧ ddcuk+1 ∧ ... ∧ dd
≤ k![
(ddch∗B)
n ∧ [
(ddcuk+1)
n ∧ ... ∧ [
(ddcun)
(by Corollary 5.6 in [Ce2])
≤ k!(e0(uk+1))
n ...(e0(un))
n .[Cn(B)]
≤ constants.[Cn(B)]
Hence
ddcu1 ∧ ... ∧ dd
cun ≤ constants.[Cn(B)]
for all Borel set B ⊂ Ω.
ii) By Proposition 3.1 we have
(−ϕ)kddcu1 ∧ ... ∧ dd
un ≤ k!
(1 − u1)(dd
ϕ)k ∧ ddcuk+1 ∧ ... ∧ dd
(ddcϕ)k ∧ ddcuk+1 ∧ ... ∧ dd
cun < +∞.
Hence (−ϕ)k ∈ L1(dd
cu1 ∧ ... ∧ dd
cun) for all ϕ ∈ F(Ω). Given a ∈ Ω let r0, R0 such that
B(a, r0) ⊂⊂ Ω ⊂⊂ B(a, R0). Then
|z − a|
≤ ga(z) ≤ log
|z − a|
for all z ∈ Ω, where ga denotes the Green function of Ω with pole at a. Since (−ga)
L1(dd
cu1 ∧ ... ∧ dd
cun), it follows that
B(a,r)
(−ga)
kddcu1 ∧ ... ∧ dd
cun → 0 as r → 0
Hence
(log r0 − log r)
B(a,r)
ddcu1 ∧ ... ∧ dd
cun ≤
B(a,r)
(−ga)
kddcu1 ∧ ... ∧ dd
cun → 0
as r → 0. This means that
B(a,r)
ddcu1 ∧ ... ∧ dd
cun = o((
log r0 − log r
)k) as r → 0
Combining this with the inequality
Cn(B(a, r),Ω) ≥ Cn(B(a, r), B(a, R0)) = (
logR0 − log r
)n = O((
log r0 − log r)n
we get
B(a,r)
ddcu1 ∧ ... ∧ dd
cun = o((Cn(B(a, r)))
The next result should be compared with Theorem 6.3 in [Ce1]
4.5. Theorem. Let u1, ..., un ∈ E . Then there exists ũ ∈ E
a such that
ddcu1 ∧ ... ∧ dd
cun = (dd
cũ)n + ddcu1 ∧ ... ∧ dd
cun|{u1=...=un=−∞}.
Proof. First, we write
ddcu1 ∧ ... ∧ dd
cun = µ + dd
cu1 ∧ ... ∧ dd
cun|{u1=...=un=−∞}.
where
µ = ddcu1 ∧ ... ∧ dd
cun|{u1>−∞}∪...∪{un>−∞}.
It is easy to see that µ ≪ Cn in every E ⊂⊂ Ω. Indeed, by Theorem 4.1 we have
ddcu1 ∧ ... ∧ dd
cun|{u1>−j} = dd
c max(u1,−j) ∧ ... ∧ dd
cun|{u1>−j}.
Hence, by Proposition 4.4 (i) it follows that ddcu1 ∧ ... ∧ dd
cun|{u1>−j} ≪ Cn in every
E ⊂⊂ Ω. Next, it remains to show that there exists ũ ∈ Ea such that µ = (ddcũ)n. Let
{Ωj} be an increasing exhaustion sequence of Ω. For each j ≥ 1 put µj = µ|Ωj . By [Åh]
there exists ũj ∈ F such that (dd
cũj)
n = µj . Notice that µj ր µ and
(ddcũj)
n ≤ µ ≤ (ddc(u1 + ... + un))
Applying the comparison principle we obtain
ũj ց ũ ≥ u1 + ... + un ∈ E .
Hence, ũ ∈ Ea and (ddcũ)n = lim
(ddcũj)
n = µ. The proof is thereby completed.
4.6. Corollary. u1, ..., un ∈ E . Then the following are equivalent
i) ddcu1 ∧ ... ∧ dd
cun ≪ Cn in every E ⊂⊂ Ω.
{u1=...=un=−∞}
ddcu1 ∧ ... ∧ dd
cun = 0.
{u1<−s,...,un<−s}∩E
ddcu1 ∧ ... ∧ dd
cun → 0 as s → +∞ for all E ⊂⊂ Ω.
Proof. Direct application of Theorem 4.5.
The comparison principle for class F was studied in [Ce3] and [H1]. By using Proposition
3.1 and Theorem 4.1 we prove a Xing type comparison principle for F
4.7. Theorem. Let u ∈ F , v ∈ E and 1 ≤ k ≤ n. Then
{u<v}
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
{u<v}
(r − w1)(dd
cv)k ∧ ddcwk+1 ∧ ... ∧ dd
{u<v}∪{u=v=−∞}
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
for all wj ∈ PSH(Ω), 0 ≤ wj ≤ 1, j = 1, ..., k, wk+1, ..., wn ∈ F and all r ≥ 1.
Proof. Let ǫ > 0. We set ṽ = max(u, v − ǫ). By a) in Proposition 3.1 we have
(ṽ − u)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cṽ)k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
Since {u < ṽ} = {u < v − ǫ} and Theorem 4.1 we have
{u<v−ǫ}
(v− ǫ−u)kddcw1 ∧ ...∧ dd
cwn +
{u≤v−ǫ}
(r−w1)(dd
cv)k ∧ ddcwk+1 ∧ ...∧ dd
{u≤v−ǫ}
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
{u<v}∪{u=v=−∞}
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
Letting ǫ ց 0 we obtain
{u<v}
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
{u<v}
(r − w1)(dd
cv)k ∧ ddcwk+1 ∧ ... ∧ dd
{u<v}∪{u=v=−∞}
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
4.8. Corollary. Let u ∈ Ea such that u ≥ v for all functions v ∈ E satisfying (ddcu)n ≤
(ddcv)n. Then
{u<v}
(v − u)nddcw1 ∧ ... ∧ dd
{u<v}
(r − w1)(dd
{u<v}
(r − w1)(dd
for all v ∈ E , r ≥ 1 and all w1, ..., wn ∈ PSH(Ω), 0 ≤ w1, ..., wn ≤ 1.
Proof. Let {Ωj} be an increasing exhaustion sequence of relatively compact subdomains
of Ω. Set µj = 1Ωj1{u>−j}(dd
cu)n, where 1E denotes the characteristic function of E ⊂ Ω.
Applying Theorem 4.1 we have
µj = 1Ωj1{u>−j}(dd
c max(u,−j))n ≤ 1Ωj (dd
c max(u,−j))n.
Take φ ∈ E0(Ω) ∩ C(Ω̄). Put
φj = max(u,−j, ajφ)
where aj =
. Then φj = max(u,−j) on Ωj+1, φj ∈ E0 and
µj ≤ 1Ωj (dd
c max(u,−j))n = 1Ωj (dd
n ≤ (ddcφj)
By Ko lodziej’s theorem (see [Ko]) there exists uj ∈ E0 such that
(ddcuj)
n = µj = 1Ωj1{u>−j}(dd
cu)n, ∀ j ≥ 1.
for all j ≥ 1. By the comparison principle we have uj ց ũ ≥ u. On the other hand, since
(ddcu)n({u = −∞}) = 0, it follows that
(ddcuj)
n = 1Ωj1{u>−j}(dd
cu)n → (ddcu)n
weakly as j → ∞. Thus (ddcũ)n = lim
(ddcuj)
n = (ddcu)n. By the hypothesis we have
ũ = u. Applying Theorem 4.7 we get
{uj<v}
(v − uj)
nddcw1 ∧ ... ∧ dd
cwn +
{uj<v}
(r − w1)(dd
{uj<v}
(r − w1)(dd
{uj<v}
(r − w1)(dd
cu)n.
Letting j → ∞ we obtain
{u<v}
(v − u)nddcw1 ∧ ... ∧ dd
cwn +
{u<v}
(r − w1)(dd
Arguing as in Theorem 4.7 we prove a Xing type comparison principle for E .
4.9. Theorem. Let u, v ∈ E and 1 ≤ k ≤ n such that lim
[u(z) − v(z)] ≥ 0. Then
{u<v}
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
{u<v}
(r − w1)(dd
cv)k ∧ ddcwk+1 ∧ ... ∧ dd
{u<v}∪{u=v=−∞}
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
for all wj ∈ PSH(Ω), 0 ≤ wj ≤ 1, j = 1, ..., k, wk+1, ..., wn ∈ E and all r ≥ 1.
Proof. Let ǫ > 0. We set ṽ = max(u, v − ǫ). By b) in Proposition 3.1 we have
(ṽ − u)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cṽ)k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
u)k ∧ ddcwk+1 ∧ ... ∧ dd
Since {u < ṽ} = {u < v − ǫ} and Theorem 4.1 we have
{u<v−ǫ}
(v− ǫ−u)kddcw1 ∧ ...∧ dd
cwn +
{u≤v−ǫ}
(r−w1)(dd
cv)k ∧ ddcwk+1 ∧ ...∧ dd
{u≤v−ǫ}
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
{u<v}∪{u=v=−∞}
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
Letting ǫ ց 0 we obtain
{u<v}
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
{u<v}
(r − w1)(dd
cv)k ∧ ddcwk+1 ∧ ... ∧ dd
{u<v}∪{u=v=−∞}
(r − w1)(dd
u)k ∧ ddcwk+1 ∧ ... ∧ dd
References
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[Xi1] Y. Xing, Continuity of the complex Monge-Ampère operator. Proc. of Amer. Math.
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Department of Mathematics
Hanoi University of Education (Dai hoc Su Pham Hanoi).
Cau giay, Ha Noi, VietNam
E-mail: phhiep−vn@yahoo.com

  In this article we will first prove a result about convergence in capacity.
Using the achieved result we will obtain a general decompositon theorem for
complex Monge-Ampere measues which will be used to prove a comparison principle
for the complex Monge-Ampere operator.

Introduction
Let Ω be a bounded hyperconvex domain in Cn. By PSH(Ω) we denote the set of plurisub-
harmonic (psh) functions on Ω. In [BT 1,2] the authors established and used the compari-
son principle to study the Dirichlet problem in PSH∩L∞loc(Ω). Recently, Cegrell introduced
a general class E of psh functions on which the complex Monge-Ampère operator (ddc.)n
can be defined. He obtained many important results of pluripotential theory in the class E .
For example, the ones on the comparison principle and solvability of the Dirichlet problem
(see [Ce 1-3]).
The main result of our paper are Theorem 4.1 and some Xing type comparision principles.
Theorem 4.1 is generalize Lemma 5.4 in [Ce1], Lemma 7.2 in [Åh] and Lemma 3.4 in [Ce3].
For definitions of Cegrell’s classes see Section 2. After giving some preliminaries, we start in
Proposition 3.1 with a comparison principle, which is analogous to a comparison principle
due to Xing (Lemma 1 in [Xi1]). It should be observed that our proof is quite different
from Xing’s proof, and the inequality we obtain is slightly stronger than Xing’s inequality,
even in the case of bounded psh functions. Using Proposition 3.1, we give in Theorem
3.5 a sufficient condition for Cn-capacity convergence of a sequence of psh functions in
the class F . This result should be compared to Theorem 3 of [Xi1] where the situation
of bounded psh functions was studied. Applying Theorem 3.5 we give generalizations of
recent results in [Cz] and [CLP] about convergences of multipole Green functions and a
criterion for pluripolarity, respectively. Section 4 focuses on Theorem 4.1 and Theorem
4.9. By applying Theorem 4.1 we give some results on class Cegrell’s classes. We prove in
Proposition 4.4 a local estimate for the Monge-Ampère measure in terms of the Beford-
Taylor relative capacity. As an application, we give in Theorem 4.5 a decomposition result
for Monge-Ampère measure, which is similar in spirit to Theorem 6.3 in [Ce1]. From
Proposition 3.1 and Theorem 4.1 we obtain easily a Xing type comparison principle for
functions in classes F and E .
Acknowledgment. We are grateful to Professor Urban Cegrell for useful discussions that
helped to improve the paper. We are grateful to Per Åhag for fruitful comments. This
work is supported by the National Research Program for Natural Sciences, Vietnam.
http://arxiv.org/abs/0704.0359v1
2. Preliminaries
First we recall some elements of pluripotential theory that will be used throughout the
paper. All this can be found in [BT2], [Ce1], [Ce2], [Le].
2.1. We will always denote by Ω a bounded hyperconvex domain in Cn unless other wise
stated. The Cn-capacity in the sense of Bedford and Taylor on Ω is the set function given
Cn(E) = Cn(E,Ω) = sup{
(ddcu)n : u ∈ PSH(Ω), −1 ≤ u ≤ 0}
for every Borel set E in Ω. It is proved in [BT2] that
Cn(E) =
(ddch∗E,Ω)
where h∗E,Ω is the upper regularization of the relative extremal function hE,Ω for E (relative
to Ω) i.e.,
hE,Ω(z) = sup{u(z) : u ∈ PSH
−(Ω), u ≤ −1 on E}.
The following concepts are taken from [Xi1] and [Xi2]
∗A sequence of functions uj on Ω is said to converge to a function u in Cn-capacity on a
set E ⊂ Ω if for every δ > 0 we have Cn({z ∈ E : |uj(z) − u(z)| > δ}) → 0 as j → ∞.
∗A family of positive measures {µα} on Ω is called uniformly absolutely continuous with
respect to Cn-capacity in a set E ⊂ Ω if for every ǫ > 0 there exists δ > 0 such that for
each Borel subset F ⊂ E with Cn(F)< δ the inequality µα(F)< ǫ holds for all α. We write
µα ≪ Cn in E uniformly for α.
2.2. The following classes of psh functions were introduced by Cegrell in [Ce1] and [Ce2]
E0 = E0(Ω) = {ϕ ∈ PSH
−(Ω) ∩ L∞(Ω) : lim
ϕ(z) = 0,
(ddcϕ)n < +∞},
F = F(Ω) = {ϕ ∈ PSH−(Ω) : ∃ E0(Ω) ∋ ϕj ց ϕ, sup
(ddcϕj)
n < +∞},
E = E(Ω) = {ϕ ∈ PSH−(Ω) : ∃ ϕK ∈ F(Ω) such that ϕK = ϕ on K, ∀K ⊂⊂ Ω},
Ea = Ea(Ω) = {u ∈ E(Ω) : (ddcu)n(E) = 0 ∀ E is pluripolar in Ω}.
For each u ∈ F(Ω), we set
e0(u) =
(ddcu)n.
2.3. Let A = {(wj , νj)}j=1,...,p be a finite subset of Ω × R
+. According to Lelong (see
[Le]), the pluricomplex Green function with poles in A is defined by
g(A)(z) = sup{u(z) : u ∈ LA}
where
LA = {u ∈ PSH
−(Ω) : u(z) − νj log |z − wj | ≤ O(1) as z → wj , j = 1, ..., p}
ν(A) =
νnj , Â = {wj}j=1,...,p.
2.4. We write lim
[u(z) − v(z)] ≥ a if for every ǫ > 0 there exists a compact set K in Ω
such that
u(z) − v(z) ≥ a− ǫ for z ∈ (Ω\K) ∩ {u > −∞}
v(z) = −∞ for z ∈ (Ω\K) ∩ {u = −∞}.
2.5. Xing’s comparison principle (see Lemma 1 in [Xi1]). Let Ω be a bounded open subset
in Cn and u, v ∈ PSH∩L∞(Ω) satisfy lim
[u(z)− v(z)] ≥ 0. Then for any constant r ≥ 1
and all wj ∈ PSH(Ω) with 0 ≤ wj ≤ 1, j = 1, 2, ..., n we have
(n!)2
{u<v}
(v − u)nddcw1 ∧ ... ∧ dd
{u<v}
(r − w1)(dd
v)n ≤
{u<v}
(r − w1)(dd
3. Some convergence theorems
In order to study the convergence of a sequence of psh functions in Cn-capacity, we start
with the following.
3.1. Proposition. a) Let u, v ∈ Fsuch that u ≤ v on Ω. Then for 1 ≤ k ≤ n
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cv)k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
for all wj ∈ PSH(Ω), 0 ≤ wj ≤ 1, j = 1, ..., k, wk+1, ..., wn ∈ F and all r ≥ 1.
b) Let u, v ∈ E such that u ≤ v on Ω and u = v on Ω\K for some K ⊂⊂ Ω. Then for
1 ≤ k ≤ n
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cv)k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
for all wj ∈ PSH(Ω), 0 ≤ wj ≤ 1, j = 1, ..., k, wk+1, ..., wn ∈ E and all r ≥ 1.
We proceed through some lemmas.
3.2. Lemma. Let u, v ∈ PSH ∩ L∞(Ω) such that u ≤ v on Ω and lim
[u(z) − v(z)] = 0.
(v − u)kddcw ∧ T ≤ k
(1 − w)(v − u)k−1ddcu ∧ T
for all w ∈ PSH(Ω), 0 ≤ w ≤ 1 and all positive closed currents T .
Proof. First, assume u, v ∈ PSH∩L∞(Ω), u ≤ v on Ω and u = v on Ω\K, K ⊂⊂ Ω. Then,
using the Stokes formula we obtain
(v − u)kddcw ∧ T =
(v − u)kddc(w − 1) ∧ T
(w − 1)ddc(v − u)k ∧ T
= −k(k − 1)
(1 − w)d(v − u) ∧ dc(v − u) ∧ T
(1 − w)(v − u)k−1ddc(u− v) ∧ T
(1 − w)(v − u)k−1ddc(u− v) ∧ T
(1 − w)(v − u)k−1ddcu ∧ T.
General case, for each ǫ > 0 we set vǫ = max(u, v − ǫ). Then vǫ ր v on Ω, vǫ ≥ u on Ω
and vǫ = u on Ω\K for some K ⊂⊂ Ω. Hence
(vǫ − u)
kddcw ∧ T ≤ k
(1 − w)(vǫ − u)
k−1ddcu ∧ T.
Since 0 ≤ vǫ − u ր v − u as ǫ ց 0, letting ǫ ց 0 we get
(v − u)kddcw ∧ T ≤ k
(1 − w)(v − u)k−1ddcu ∧ T.
3.3. Lemma. Let u, v ∈ PSH ∩ L∞(Ω) such that u ≤ v on Ω and lim
[u(z) − v(z)] = 0.
Then for 1 ≤ k ≤ n
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cv)k ∧ T
(r − w1)(dd
cu)k ∧ T.
for all w1, ..., wk ∈ PSH(Ω), 0 ≤ wj ≤ 1 ∀ j = 1, ..., k, wk+1, ..., wn ∈ E and all r ≥ 1.
Proof. To simplify the notation we set
T = ddcwk+1 ∧ ... ∧ dd
First, assume that u, v ∈ PSH ∩ L∞(Ω), u ≤ v on Ω, and u = v on Ω\K, K ⊂⊂ Ω. Using
Lemma 3.2 we get
(v − u)kddcw1 ∧ ... ∧ dd
cwn ≤ k
(v − u)k−1ddcw1 ∧ ... ∧ dd
cwk−1 ∧ dd
cu ∧ T
≤ ...
(v − u)ddcw1 ∧ (dd
cu)k−1 ∧ T
(v − u)ddcw1 ∧ [
(ddcu)i ∧ (ddcv)k−i−1] ∧ T
(w1 − r)dd
c(v − u) ∧ [
(ddcu)i ∧ (ddcv)k−i−1] ∧ T
(r − w1)dd
c(u− v) ∧ [
(ddcu)i ∧ (ddcv)k−i−1] ∧ T
(r − w1)[(dd
cu)k − (ddcv)k] ∧ T.
General case, for each ǫ > 0 we put vǫ = max(u, v − ǫ). Then vǫ ր v on Ω, vǫ ≥ u on Ω
and vǫ = u on Ω\K for some K ⊂⊂ Ω. Hence
(vǫ − u)
kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
k ∧ T
(r − w1)(dd
u)k ∧ T.
Observe that 0 ≤ vǫ − u ր v− u and (dd
k ∧ T → (ddcv)k ∧ T weakly as ǫ ց 0, r−w1
is lower semicontinuous, by letting ǫ ց 0 we have
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cv)k ∧ T
(r − w1)(dd
cu)k ∧ T.
The proof is finished.
Proof of Proposition 3.1. a) Let E0 ∋ uj ց u and E0 ∋ vj ց v as in the definition of F .
Replace vj by max(uj , vj) we may assume that uj ≤ vj for j ≥ 1. By Lemma 3.3 we have
(vj − ut)
kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
k ∧ ddcwk+1 ∧ ... ∧ dd
for t ≥ j ≥ 1. By Proposition 5.1 in [Ce2] letting t → ∞ in the above inequality we have
(vj − u)
w1 ∧ ... ∧ dd
(r − w1)(dd
k ∧ T
(r − w1)(dd
cu)k ∧ T
for j ≥ 1. Next letting j → ∞ again by Proposition 5.1 in [Ce2] we get the desired
conclusion.
b) Let G,W be open sets such that K ⊂⊂ G ⊂⊂ W ⊂⊂ Ω. According to the remark
following Definition 4.6 in [Ce2] we can choose a function ṽ ∈ F such that ṽ ≥ v and ṽ = v
on W . Set
u on G
ṽ on Ω\G
Since u = v = ṽ on W\K we have ũ ∈ PSH−(Ω). It is easy to see that ũ ∈ F , ũ ≤ ṽ and
ũ = u on W . By a) we have
(ṽ − ũ)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cṽ)k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
ũ)k ∧ ddcwk+1 ∧ ... ∧ dd
Since ũ = ṽ on Ω\G we have
(ṽ − ũ)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cṽ)k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
cũ)k ∧ ddcwk+1 ∧ ... ∧ dd
Since ũ = u, ṽ = v on W and u = v on Ω\K we obtain
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cv)k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
3.4. Proposition. Let u, v ∈ F and u ≤ v on Ω. Then
(v − u)nddcw1 ∧ ... ∧ dd
(−w1)[(dd
u)n − (ddcv)n]
for all wj ∈ PSH(Ω), −1 ≤ wj ≤ 0, j = 1, ..., n.
Proof. The proposition follows from Proposition 3.1 with k = n, r = 1 and wj are replaced
by wj + 1.
3.5. Theorem. Let u, uj ∈ F and uj ≤ u for j ≥ 1. Assume that sup
(ddcuj)
n < +∞
and ||(ddcuj)
n − (ddcu)n||E → 0 as j → ∞ for all E ⊂⊂ Ω. Then uj → u in Cn-capacity
on every E ⊂⊂ Ω as j → ∞.
Proof. Let Ω′ ⊂⊂ Ω and δ > 0. Put
Aj = {z ∈ Ω′ : |uj − u| ≥ δ} = {z ∈ Ω′ : u− uj ≥ δ}.
We prove that Cn(Aj) → 0 as j → ∞. Given ǫ > 0. By quasicontinuity of u and uj , there
is an open set G in Ω such that Cn(G) < ǫ, and uj |Ω\G, u|Ω\G are continuous. We have
Aj = Bj ∪ {z ∈ G : u− uj ≥ δ}.
where Bj = {z ∈ Ω′\G : u− uj ≥ δ} are compact sets in Ω and
Cn(Aj) ≤ lim
Cn(Bj) + ǫ
We claim that lim
Cn(Bj) = 0. By Proposition 3.4 we have
Cn(Bj) =
(ddch∗Bj )
(u− uj)
n(ddch∗Bj )
(−h∗Bj )[(dd
n − (ddcu)n]
{||(ddcuj)
n − (ddcu)n||K +
(−hΩ′)[(dd
n + (ddcu)n]}
{||(ddcuj)
n − (ddcu)n||K + sup
|hΩ′ |[sup
(ddcuj)
(ddcu)n]}.
As lim
hΩ′(z) = 0 there exists K ⊂⊂ Ω such that
|hΩ′ |[sup
(ddcuj)
(ddcu)n] < ǫ.
By the hypothesis
||(ddcuj)
n − (ddcu)n||K < ǫ for j > j0.
Cn(Bj) < 2ǫ for j > j0.
This proves the claim and hence the theorem.
As an application of Theorem 3.5 we have the following
3.6. Proposition. Let g(Aj) be multipolar Green functions on Ω such that
Âj = {w
1, ..., w
} → ∂Ω and sup
ν(Aj) = sup
)n < +∞
Then g(Aj) → 0 as j → ∞ in Cn-capacity.
Proof. By the hypothesis we have
(ddcg(Aj))
n(Ω) = sup
ν(Aj) < +∞
||(ddcg(Aj))
n||K → 0 as j → ∞ for all K ⊂⊂ Ω.
Theorem 3.5 implies that g(Aj) → 0 as j → ∞ in Cn-capacity.
This section ends up with a criterion for pluripolarity
3.7. Theorem. Let uj ∈ F such that sup
(ddcuj)
n < +∞.
Then there is a constant A > 0 such that
i)( lim
∗ ∈ F .
ii)Cn({z ∈ Ω : ( lim
∗(z) < −t}) ≤ A
iii){z ∈ Ω : lim
uj(z) = −∞} is pluripolar.
Proof. i) For each j ≥ 1 put vj = sup{uj, uj+1, ...}. By [Ce2] v
j ∈ F and
(ddcv∗j )
n ≤ sup
(ddcuj)
n < +∞.
By [Ce2] we have v∗j ց v ∈ F .
ii) By Proposition 3.1 in [CKZ] we have
Cn{z ∈ Ω : ( lim
∗(z) < −t} = Cn{z ∈ Ω : v(z) < −t} ≤
2ne0(v)
where A = 2ne0(v).
iii) According to [BT2] we have
Cn{z ∈ Ω : lim
uj(z) = −∞} = Cn{z ∈ Ω : v(z) = −∞} = 0.
Remark. Theorem 3.7 in the case where uj are multipole Green functions was proved by
D.Coman, N.Levenberg and A.Poletsky in Theorem 4.1 of [CLP].
4. Some properties of the Cegrell’s classes and applications
In this section, first we prove the following
4.1. Theorem. Let u, u1, ..., un−1 ∈ E , v ∈ PSH
−(Ω) and T = ddcu1 ∧ ... ∧ dd
cun−1.
ddc max(u, v) ∧ T |{u>v} = dd
cu ∧ T |{u>v}.
We need the following well-known fact.
4.2. Lemma. Let µ be a measure on Ω and f : Ω → R a measurable function on Ω. The
following are equivalent
i)µ(E) = 0 for all Borell sets E ⊂ {f 6= 0}.
fdµ = 0 for every measurable set E in Ω.
Proof. i)⇒ii) follows from:
fdµ =
E\{f=0}
fdµ +
E∩{f=0}
fdµ = 0
ii)⇒i). It suffices to show that µ = 0 on every Xδ = {f > δ > 0}. By the Hahn
decomposition theorem, there exist measurable subsets X+
and X−
of Xδ such that Xδ =
= ∅ and µ ≥ 0 on X+
, µ ≤ 0 on X−
. We have
δµ(X+
fdµ = 0
δµ(X−
fdµ = 0
Hence, µ(X+
) = µ(X−
) = 0. Therefore, we have µ = 0 on Xδ.
Proof of Theorem 4.1.
a) First we prove the proposition for v ≡ a < 0. According to the remark following
Definition 4.6 in [Ce2], without loss of generality we may assume that u, u1, ..., un−1 ∈ F .
Using Theorem 2.1 in [Ce2] we can find
E0 ∩ C(Ω̄) ∋ u
j ց u, E0 ∩ C(Ω̄) ∋ u
ց uk, k = 1, ..., n− 1.
Since {uj > a} is open we have
ddc max(uj, a) ∧ Tj |{uj>a} = dd
cuj ∧ Tj |{uj>a}.
Thus from the inclusion {u > a} ⊂ {uj > a} we obtain
ddc max(uj , a) ∧ Tj |{u>a} = dd
cuj ∧ Tj |{u>a}.
where Tj = dd
1 ∧ ... ∧ dd
n−1. By Corollary 5.2 in [Ce2], it follows that
max(u− a, 0)ddc max(uj , a) ∧ Tj → max(u− a, 0)dd
c max(u, a) ∧ T.
max(u− a, 0)ddcuj ∧ Tj → max(u− a, 0)dd
cu ∧ T.
Hence
max(u− a, 0)[ddc max(u, a) ∧ T − ddcu ∧ T ] = 0.
Using Lemma 4.2 we have
ddc max(u, a) ∧ T = ddcu ∧ T on {u > a}.
b) Assume that v ∈ PSH−(Ω). Since {u > v} =
{u > a > v}, it suffices to show
ddc max(u, v) ∧ T = ddcu ∧ T on {u > a > v}
for all a ∈ Q−. Since max(u, v) ∈ E , by a) we have
ddc max(u, v) ∧ T |{max(u,v)>a} = dd
c max(max(u, v), a) ∧ T |{max(u,v)>a}
= ddc max(u, v, a) ∧ T |{max(u,v)>a}.
(2) ddcu ∧ T |{u>a} = dd
c max(u, a) ∧ T |{u>a}.
Since max(u, v, a) = max(u, a) on set open {a > v} , we have
(3) ddc max(u, v, a) ∧ T |{a>v} = dd
c max(u, a) ∧ T |{a>v}.
Since {u > a > v} ⊂ {u > a}, {a > v}, {max(u, v) > a} and (1), (2), (3) we have
ddc max(u, v) ∧ T |{u>a>v} = dd
cu ∧ T |{u>a>v}.
The next result is an analogue of an inequality due to Demaily in [De2]
4.3. Proposition. a) u, v ∈ E such that (ddcu)n({u = v = −∞}) = 0. Then
(ddc max(u, v))n ≥ 1{u≥v}(dd
cu)n + 1{u<v}(dd
where 1E denotes the characteristic function of E.
b) Let µ be a positive measure which vanishes on all pluripolar subsets of Ω. Suppose
u, v ∈ E such that (ddcu)n ≥ µ, (ddcv)n ≥ µ. Then (ddc max(u, v))n ≥ µ.
Proof. a) For each ǫ > 0 put Aǫ = {u = v − ǫ}\{u = v = −∞}. Since Aǫ ∩ Aδ = ∅ for
ǫ 6= δ there exists ǫj ց 0 such that (dd
cu)n(Aǫj ) = 0 for j ≥ 1. On the other hand, since
(ddcu)n({u = v = −∞}) = 0 we have (ddcu)n({u = v− ǫj}) = 0 for j ≥ 1. Since Theorem
4.1 it follows that
(ddc max(u, v − ǫj))
n ≥ (ddc max(u, v − ǫj))
n|{u>v−ǫj} + (dd
c max(u, v − ǫj))
n|{u<v−ǫj}
= (ddcu)n|{u≥v−ǫj} + (dd
cv)n|{u<v−ǫj}
= 1{u≥v−ǫj}(dd
cu)n + 1{u<v−ǫj}(dd
≥ 1{u≥v}(dd
cu)n + 1{u<v−ǫj}(dd
cv)n.
Letting j → ∞ and by Remark under Theorem 5.15 in [Ce2] we get
(ddc max(u, v))n ≥ 1{u≥v}(dd
cu)n + 1{u<v}(dd
because max(u, v − ǫj) ր max(u, v) and 1{u<v−ǫj} ր 1{u<v} as j → ∞.
b) Argument as a)
4.4. Proposition. Let u1, ..., uk ∈ PSH(Ω) ∩ L
∞(Ω) and uk+1, ..., un ∈ E . Then
ddcu1 ∧ ... ∧ dd
cun = O((Cn(B))
n ) for all Borel sets B ⊂ Ω′ ⊂⊂ Ω.
B(a,r)
ddcu1 ∧ ... ∧ dd
cun = o((Cn(B(a, r)))
n ) as r → 0 for all a ∈ Ω.
where B(a, r) = {z ⊂ Cn : |z − a| < r}
Proof. We may assume that 0 ≤ uj ≤ 1 for j = 1, ..., k. On the other hand, by the remark
following Defintion 4.6 in [Ce2] we again may assume that uk+1, ..., un ∈ F .
i) For each open set B ⊂⊂ Ω, applying Proposition 3.1 we get
ddcu1 ∧ ... ∧ dd
cun =
(−h∗B)
kddcu1 ∧ ... ∧ dd
(−h∗B)
kddcu1 ∧ ... ∧ dd
(1 − u1)(dd
ch∗B)
k ∧ ddcuk+1 ∧ ... ∧ dd
(ddch∗B)
k ∧ ddcuk+1 ∧ ... ∧ dd
≤ k![
(ddch∗B)
n ∧ [
(ddcuk+1)
n ∧ ... ∧ [
(ddcun)
(by Corollary 5.6 in [Ce2])
≤ k!(e0(uk+1))
n ...(e0(un))
n .[Cn(B)]
≤ constants.[Cn(B)]
Hence
ddcu1 ∧ ... ∧ dd
cun ≤ constants.[Cn(B)]
for all Borel set B ⊂ Ω.
ii) By Proposition 3.1 we have
(−ϕ)kddcu1 ∧ ... ∧ dd
un ≤ k!
(1 − u1)(dd
ϕ)k ∧ ddcuk+1 ∧ ... ∧ dd
(ddcϕ)k ∧ ddcuk+1 ∧ ... ∧ dd
cun < +∞.
Hence (−ϕ)k ∈ L1(dd
cu1 ∧ ... ∧ dd
cun) for all ϕ ∈ F(Ω). Given a ∈ Ω let r0, R0 such that
B(a, r0) ⊂⊂ Ω ⊂⊂ B(a, R0). Then
|z − a|
≤ ga(z) ≤ log
|z − a|
for all z ∈ Ω, where ga denotes the Green function of Ω with pole at a. Since (−ga)
L1(dd
cu1 ∧ ... ∧ dd
cun), it follows that
B(a,r)
(−ga)
kddcu1 ∧ ... ∧ dd
cun → 0 as r → 0
Hence
(log r0 − log r)
B(a,r)
ddcu1 ∧ ... ∧ dd
cun ≤
B(a,r)
(−ga)
kddcu1 ∧ ... ∧ dd
cun → 0
as r → 0. This means that
B(a,r)
ddcu1 ∧ ... ∧ dd
cun = o((
log r0 − log r
)k) as r → 0
Combining this with the inequality
Cn(B(a, r),Ω) ≥ Cn(B(a, r), B(a, R0)) = (
logR0 − log r
)n = O((
log r0 − log r)n
we get
B(a,r)
ddcu1 ∧ ... ∧ dd
cun = o((Cn(B(a, r)))
The next result should be compared with Theorem 6.3 in [Ce1]
4.5. Theorem. Let u1, ..., un ∈ E . Then there exists ũ ∈ E
a such that
ddcu1 ∧ ... ∧ dd
cun = (dd
cũ)n + ddcu1 ∧ ... ∧ dd
cun|{u1=...=un=−∞}.
Proof. First, we write
ddcu1 ∧ ... ∧ dd
cun = µ + dd
cu1 ∧ ... ∧ dd
cun|{u1=...=un=−∞}.
where
µ = ddcu1 ∧ ... ∧ dd
cun|{u1>−∞}∪...∪{un>−∞}.
It is easy to see that µ ≪ Cn in every E ⊂⊂ Ω. Indeed, by Theorem 4.1 we have
ddcu1 ∧ ... ∧ dd
cun|{u1>−j} = dd
c max(u1,−j) ∧ ... ∧ dd
cun|{u1>−j}.
Hence, by Proposition 4.4 (i) it follows that ddcu1 ∧ ... ∧ dd
cun|{u1>−j} ≪ Cn in every
E ⊂⊂ Ω. Next, it remains to show that there exists ũ ∈ Ea such that µ = (ddcũ)n. Let
{Ωj} be an increasing exhaustion sequence of Ω. For each j ≥ 1 put µj = µ|Ωj . By [Åh]
there exists ũj ∈ F such that (dd
cũj)
n = µj . Notice that µj ր µ and
(ddcũj)
n ≤ µ ≤ (ddc(u1 + ... + un))
Applying the comparison principle we obtain
ũj ց ũ ≥ u1 + ... + un ∈ E .
Hence, ũ ∈ Ea and (ddcũ)n = lim
(ddcũj)
n = µ. The proof is thereby completed.
4.6. Corollary. u1, ..., un ∈ E . Then the following are equivalent
i) ddcu1 ∧ ... ∧ dd
cun ≪ Cn in every E ⊂⊂ Ω.
{u1=...=un=−∞}
ddcu1 ∧ ... ∧ dd
cun = 0.
{u1<−s,...,un<−s}∩E
ddcu1 ∧ ... ∧ dd
cun → 0 as s → +∞ for all E ⊂⊂ Ω.
Proof. Direct application of Theorem 4.5.
The comparison principle for class F was studied in [Ce3] and [H1]. By using Proposition
3.1 and Theorem 4.1 we prove a Xing type comparison principle for F
4.7. Theorem. Let u ∈ F , v ∈ E and 1 ≤ k ≤ n. Then
{u<v}
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
{u<v}
(r − w1)(dd
cv)k ∧ ddcwk+1 ∧ ... ∧ dd
{u<v}∪{u=v=−∞}
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
for all wj ∈ PSH(Ω), 0 ≤ wj ≤ 1, j = 1, ..., k, wk+1, ..., wn ∈ F and all r ≥ 1.
Proof. Let ǫ > 0. We set ṽ = max(u, v − ǫ). By a) in Proposition 3.1 we have
(ṽ − u)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cṽ)k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
Since {u < ṽ} = {u < v − ǫ} and Theorem 4.1 we have
{u<v−ǫ}
(v− ǫ−u)kddcw1 ∧ ...∧ dd
cwn +
{u≤v−ǫ}
(r−w1)(dd
cv)k ∧ ddcwk+1 ∧ ...∧ dd
{u≤v−ǫ}
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
{u<v}∪{u=v=−∞}
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
Letting ǫ ց 0 we obtain
{u<v}
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
{u<v}
(r − w1)(dd
cv)k ∧ ddcwk+1 ∧ ... ∧ dd
{u<v}∪{u=v=−∞}
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
4.8. Corollary. Let u ∈ Ea such that u ≥ v for all functions v ∈ E satisfying (ddcu)n ≤
(ddcv)n. Then
{u<v}
(v − u)nddcw1 ∧ ... ∧ dd
{u<v}
(r − w1)(dd
{u<v}
(r − w1)(dd
for all v ∈ E , r ≥ 1 and all w1, ..., wn ∈ PSH(Ω), 0 ≤ w1, ..., wn ≤ 1.
Proof. Let {Ωj} be an increasing exhaustion sequence of relatively compact subdomains
of Ω. Set µj = 1Ωj1{u>−j}(dd
cu)n, where 1E denotes the characteristic function of E ⊂ Ω.
Applying Theorem 4.1 we have
µj = 1Ωj1{u>−j}(dd
c max(u,−j))n ≤ 1Ωj (dd
c max(u,−j))n.
Take φ ∈ E0(Ω) ∩ C(Ω̄). Put
φj = max(u,−j, ajφ)
where aj =
. Then φj = max(u,−j) on Ωj+1, φj ∈ E0 and
µj ≤ 1Ωj (dd
c max(u,−j))n = 1Ωj (dd
n ≤ (ddcφj)
By Ko lodziej’s theorem (see [Ko]) there exists uj ∈ E0 such that
(ddcuj)
n = µj = 1Ωj1{u>−j}(dd
cu)n, ∀ j ≥ 1.
for all j ≥ 1. By the comparison principle we have uj ց ũ ≥ u. On the other hand, since
(ddcu)n({u = −∞}) = 0, it follows that
(ddcuj)
n = 1Ωj1{u>−j}(dd
cu)n → (ddcu)n
weakly as j → ∞. Thus (ddcũ)n = lim
(ddcuj)
n = (ddcu)n. By the hypothesis we have
ũ = u. Applying Theorem 4.7 we get
{uj<v}
(v − uj)
nddcw1 ∧ ... ∧ dd
cwn +
{uj<v}
(r − w1)(dd
{uj<v}
(r − w1)(dd
{uj<v}
(r − w1)(dd
cu)n.
Letting j → ∞ we obtain
{u<v}
(v − u)nddcw1 ∧ ... ∧ dd
cwn +
{u<v}
(r − w1)(dd
Arguing as in Theorem 4.7 we prove a Xing type comparison principle for E .
4.9. Theorem. Let u, v ∈ E and 1 ≤ k ≤ n such that lim
[u(z) − v(z)] ≥ 0. Then
{u<v}
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
{u<v}
(r − w1)(dd
cv)k ∧ ddcwk+1 ∧ ... ∧ dd
{u<v}∪{u=v=−∞}
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
for all wj ∈ PSH(Ω), 0 ≤ wj ≤ 1, j = 1, ..., k, wk+1, ..., wn ∈ E and all r ≥ 1.
Proof. Let ǫ > 0. We set ṽ = max(u, v − ǫ). By b) in Proposition 3.1 we have
(ṽ − u)kddcw1 ∧ ... ∧ dd
cwn +
(r − w1)(dd
cṽ)k ∧ ddcwk+1 ∧ ... ∧ dd
(r − w1)(dd
u)k ∧ ddcwk+1 ∧ ... ∧ dd
Since {u < ṽ} = {u < v − ǫ} and Theorem 4.1 we have
{u<v−ǫ}
(v− ǫ−u)kddcw1 ∧ ...∧ dd
cwn +
{u≤v−ǫ}
(r−w1)(dd
cv)k ∧ ddcwk+1 ∧ ...∧ dd
{u≤v−ǫ}
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
{u<v}∪{u=v=−∞}
(r − w1)(dd
cu)k ∧ ddcwk+1 ∧ ... ∧ dd
Letting ǫ ց 0 we obtain
{u<v}
(v − u)kddcw1 ∧ ... ∧ dd
cwn +
{u<v}
(r − w1)(dd
cv)k ∧ ddcwk+1 ∧ ... ∧ dd
{u<v}∪{u=v=−∞}
(r − w1)(dd
u)k ∧ ddcwk+1 ∧ ... ∧ dd
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Department of Mathematics
Hanoi University of Education (Dai hoc Su Pham Hanoi).
Cau giay, Ha Noi, VietNam
E-mail: phhiep−vn@yahoo.com
