# Weighted Pluricomplex Energy II

Authors: Slimane Benelkourchi
Journal: International Journal of Partial Differential Equations
 (2015)

---

## Abstract

We continue our study of the complex Monge-Ampère operator on the weighted pluricomplex energy classes. We give more characterizations of the range of the classesE
χ by the complex Monge-Ampère operator. In particular, we prove that a nonnegative Borel measure μ is the Monge-Ampère of a unique function φ
∈
E
χ if and only if χ
(
E
χ
)
⊂
L
1
(
d
μ
). Then we show that if μ
=
(
d
d
c
φ
)
n for some φ
∈
E
χ then μ
=
(
d
d
c
u
)
n for some φ
∈
E
χ, where f is given boundary data. If moreover the nonnegative Borel measure μ is suitably dominated by the Monge-Ampère capacity, we establish a priori estimates on the capacity of sublevel sets of the solutions. As a consequence, we give a priori bounds of the solution of the Dirichlet problem in the case when the measure has a density in some Orlicz space.


---

## Body

## 1. Introduction

LetΩ
⊂
C
n be a bounded hyperconvex domain, that is, a connected, bounded open subset of C
n, such that there exists a negative plurisubharmonic function ρ such that {
z
∈
Ω
;
ρ
(
z
)
<
-
c
}
⋐
Ω, for all c
>
0. Such a function ρ is called an exhaustion function. We let PSH
(
Ω
) denote the cone of plurisubharmonic functions (psh for short) on Ω and let PS
H
-
(
Ω
) denote the subclass of negative functions.As known (see [1, 2]), the complex Monge-Ampère operator (
d
d
c
·
)
n is well defined, as a nonnegative measure, on the set of locally bounded plurisubharmonic functions. Therefore the question of describing the measures which are the Monge-Ampère of bounded psh functions is very important for pluripotential theory, complex dynamic, and complex geometry. This problem has been studied extensively by various authors; see, for example, [2–6] and reference therein. In [7], Cegrell introduced the pluricomplex energy classes E
p
(
Ω
) and F
p
(
Ω
) (p
≥
1) on which the complex Monge-Ampère operator is well defined. He proved that a measure μ is the Monge-Ampère of some function u
∈
E
p
(
Ω
) if and only if it satisfies
(1)
∫
Ω
-
v
p
d
μ
≤
Const
∫
Ω
-
v
p
d
d
c
v
n
p
/
(
n
+
p
)
,
∫
Ω
-
v
p
f
g
f
g
d
d
c
v
n
∀
v
∈
E
0
(
Ω
)
,
where E
0
(
Ω
) is the cone of all bounded psh functions φ defined on the domain Ω with finite total Monge-Ampère mass and lim
⁡
z
→
ζ
φ
(
z
)
=
0, for every ζ
∈
∂
Ω. Recently, Åhag et al. in [8] proved that, in the case p
=
1, inequality (1) is equivalent to E
1
(
Ω
)
⊂
L
1
(
d
μ
). In this note, our first objective is to extend this result by showing that, for all positive number p, inequality (1) is equivalent to E
p
(
Ω
)
⊂
L
p
(
d
μ
). In fact, we prove some more general result. Given a nondecreasing function χ
:
R
-
→
R
-, we consider the set E
χ
(
Ω
) of plurisubharmonic functions of finite χ-weighted Monge-Ampère energy and, in some sense, has boundary values zero. These are the functions u
∈
PSH
(
Ω
) for which there exists a decreasing sequence u
j
∈
E
0
(
Ω
) with limit u and
(2)
sup
⁡
j
∈
N
∫
Ω
-
χ
(
u
j
)
d
d
c
u
j
n
<
∞
.Then we have the following characterization of the image of the complex Monge-Ampère acting in the classE
χ
(
Ω
).Theorem 1.
Letχ
:
R
-
→
R
- be an increasing convex or homogeneous function such that χ
(
-
∞
)
=
-
∞. The following assertions are equivalent:(1)
there exists a unique functionφ
∈
E
χ
(
Ω
) such that μ
=
(
d
d
c
φ
)
n;
(2)
χ
(
E
χ
(
Ω
)
)
⊂
L
1
(
d
μ
).Next, we extend our previous result to families of functions having prescribed boundary data. Letf
∈
PSH
(
Ω
) be a maximal psh function. We define the class E
χ
(
f
) to be the class of psh functions u such that there exists a function φ
∈
E
χ
(
Ω
) such that
(3)
φ
(
z
)
+
f
(
z
)
≤
u
(
z
)
≤
f
(
z
)
,
∀
z
∈
Ω
.
Some particular cases of the classes E
χ
(
f
) have been studied in [6, 7, 9–16].More precisely, we prove the following result.Theorem 2.
Letμ be a nonnegative measure in Ω, let χ
:
R
-
→
R
- be an increasing convex or homogeneous function such that χ
(
-
∞
)
=
-
∞, and let f be a maximal function. Then, if μ
=
(
d
d
c
u
)
n for some u
∈
E
χ
(
Ω
) then there exists a unique function φ
∈
E
χ
(
f
) such that μ
=
(
d
d
c
φ
)
n.Moreover, when the nonnegative measureμ is dominated by the Monge-Ampère capacity, we give an estimate of the growth of solutions of the equation (
d
d
c
φ
)
n
=
μ. As in [12], let us consider the function
(4)
F
μ
(
t
)
:
=
sup
⁡
μ
K
,
K
⋐
Ω
;
ca
p
Ω
K
≤
t
,
∀
t
≥
0
.
Then F
∶
=
F
μ is a nondecreasing function on R
+ and satisfies
(5)
μ
K
≤
F
μ
ca
p
μ
K
,
∀
Borel
subsets
K
⊂
Ω
.
Write F
(
x
)
=
F
ε
(
x
)
=
x
ε
(
-
ln
⁡
⁡
(
x
/
n
)
)
n where ε
:
R
+
→
R
+ is nondecreasing.Such measures dominated by the Monge-Ampère capacity have been extensively studied by Kołodziej in [3–5]. He proved that if ϕ
:
∂
Ω
→
R is a continuous function and ∫
0
+
∞
ε
(
t
)
d
t
<
+
∞, then μ is the Monge-Ampère of a unique function φ
∈
PSH
(
Ω
) with φ
/
∂
Ω
=
ϕ.When∫
0
+
∞
ε
(
s
)
d
s
=
+
∞, we have the following estimate.Theorem 3.
Letμ be a positive finite measure. Assume, for all compact subsets K
⊂
Ω,
(6)
μ
(
K
)
≤
F
ε
c
a
p
Ω
K
.
Then there exists a unique function φ
∈
F
(
f
) such that μ
=
(
d
d
c
φ
)
n, and
(7)
c
a
p
Ω
φ
<
f
-
s
≤
exp
⁡
-
n
H
-
1
s
,
∀
s
>
0
.
Here H
-
1 is the reciprocal function of H
(
x
)
=
e
∫
0
x
ε
(
t
)
d
t
+
s
0
(
μ
).
In particular if∫
0
+
∞
ε
(
t
)
d
t
<
+
∞ then
(8)
0
≤
f
-
φ
≤
e
∫
0
+
∞
ε
(
t
)
d
t
+
e
ε
(
0
)
+
μ
μ
1
/
n
.The paper is organised as follows. In Section2, we recall the definitions of the energy classes E
χ
(
Ω
) and some classes of psh functions introduced by Cegrell [7, 13, 14] and we prove Theorem 1. In Section 3, we prove Theorem 2. As a consequence, we generalize the main result in the paper [9]. In Section 4, we prove Theorem 3. As application, we give a priori bound of the solution of Dirichlet problem in the case when the measure μ
=
g
d
λ, where g belongs to some Orlicz space L
log
⁡
α
L.


## 2. Energy Classes with Zero Boundary DataE
χ

Let us recall some Cegrell’s classes (Cf. [7, 13, 14]). The class E
(
Ω
) is the set of plurisubharmonic functions u such that, for all z
0
∈
Ω, there exists a neighbourhood V
z
0 of z
0 and u
j
∈
E
0
(
Ω
), a decreasing sequence which converges towards u in V
z
0 and satisfies sup
⁡
j
∫
Ω
(
d
d
c
u
j
)
n
<
+
∞. Cegrell [13] has shown that the operator (
d
d
c
·
)
n is well defined on E
(
Ω
), continuous under decreasing limits, and the class E
(
Ω
) is stable under taking maximum; that is, if u
∈
E
(
Ω
) and v
∈
PS
H
-
(
Ω
) then max
⁡
(
u
,
v
)
∈
E
(
Ω
). This class is the largest class with these properties (Theorem 4.5 in [13]). The class E
(
Ω
) has been further characterized by Błocki [17, 18] and Le Mau et al. in [19].The classF
(
Ω
) is the global version of E
(
Ω
): a function u belongs to F
(
Ω
) if and only if there exists a decreasing sequence u
j
∈
E
0
(
Ω
) converging towards u in all of  Ω, which satisfies sup
⁡
j
∫
Ω
(
d
d
c
u
j
)
n
<
+
∞. The class E
(
Ω
) has been further characterized in [12, 17].LetΩ
j
⋐
Ω be an increasing sequence of strictly pseudoconvex domains such that Ω
=
⋃
j
Ω
j. Let u
∈
E
(
Ω
) be a given psh function and put
(9)
u
Ω
j
:
=
sup
⁡
φ
∈
PSH
Ω
;
φ
≤
u
on
⁡
Ω
∖
Ω
j
.
Then we have u
Ω
j
∈
E
(
Ω
) and u
Ω
j is an increasing sequence. Let u
~
:
=
(
lim
⁡
j
u
Ω
j
)
*. It follows from the properties of E
(
Ω
) that u
~
∈
E
(
Ω
). Note that the definition of u
~ is independent of the choice of the sequence Ω
j and is maximal; that is, (
d
d
c
u
~
)
n
=
0. u
~ is the smallest maximal psh function above u. Define N
(
Ω
)
:
=
{
u
∈
E
(
Ω
)
;
u
~
=
0
}. In fact, this class is the analogous of potentials for subharmonic functions.Definition 4.
Letχ
:
R
-
→
R
- be a nondecreasing function. We let E
χ
(
Ω
) denote the set of all functions u
∈
PSH
(
Ω
) for which there exists a sequence u
j
∈
E
0
(
Ω
) decreasing to u in Ω and satisfying
(10)
sup
⁡
j
∈
N
∫
Ω
-
χ
(
u
j
)
d
d
c
u
j
n
<
∞
.It was proved in [15, 20] that if χ
≢
0 then
(11)
E
χ
(
Ω
)
⊂
E
(
Ω
)
.
In particular, for any function u
∈
E
χ
(
Ω
), the complex Monge-Ampère operator (
d
d
c
u
)
n is well defined as nonnegative measure. Furthermore, if χ
(
-
t
)
<
0 for all t
>
0, then
(12)
E
χ
(
Ω
)
=
u
∈
N
(
Ω
)
;
∫
Ω
-
χ
(
u
)
d
d
c
u
n
<
+
∞
.
The class E
χ
(
Ω
) has been characterized by the speed of decrease of the capacity of sublevel sets [11, 12].Recall that the Monge-Ampère capacity has been introduced and studied by Bedford and Taylor in [1]. Given K
⊂
Ω, a compact subset, its Monge-Ampère capacity relatively to Ω is defined by
(13)
ca
p
Ω
(
K
)
:
=
sup
⁡
∫
K
d
d
c
u
n
;
u
∈
PSH
(
Ω
)
,
-
1
≤
u
≤
0
.
The following estimates (cf [12]) will be useful later on. For any φ
∈
E
0
(14)
t
n
ca
p
Ω
(
φ
<
-
s
-
t
)
≤
∫
(
φ
<
-
s
)
d
d
c
φ
n
≤
s
n
ca
p
Ω
(
φ
<
-
s
)
,
∀
s
,
t
>
0
.Letχ
:
R
-
→
R
- be a nondecreasing function. Without loss of generality, from now on, we assume that χ
(
0
)
=
0. We define the class E
^
χ
(
Ω
)
(15)
E
^
χ
(
Ω
)
:
=
∫
t
χ
+
∞
‍
φ
∈
PS
H
-
Ω
;
E
^
χ
Ω
≔
1
∫
t
χ
+
∞
‍
t
n
χ
′
t
ca
p
Ω
φ
<
-
t
d
t
<
+
∞
.Proposition 5.
One hasE
^
χ
(
Ω
)
⊂
E
χ
(
Ω
), while
(16)
E
χ
Ω
⊂
E
^
χ
^
Ω
,
w
h
e
r
e
χ
^
(
t
)
=
χ
t
2
.
Moreover, if χ
:
]
-
∞
,
-
t
χ
[
→
R
- is convex, then
(17)
E
χ
(
Ω
)
=
E
^
χ
(
Ω
)
.
Here t
χ denote the real number satisfying χ
(
t
)
<
0, for all t
<
-
t
χ, and χ
(
t
)
=
0, for all t
≥
-
t
χ.Proof.
Compare with [12, 20].Theorem 6.
Letχ
:
R
-
→
R
- be an increasing convex function such that χ
(
-
∞
)
=
-
∞. The following conditions are equivalent:(1)
there exists a unique functionφ
∈
E
χ
(
Ω
) such that μ
=
(
d
d
c
φ
)
n;
(2)
χ
(
E
χ
(
Ω
)
)
⊂
L
1
(
Ω
,
d
μ
);
(3)
there exists a constantC
>
0 such that
(18)
∫
Ω
-
χ
(
u
)
d
μ
≤
C
,
∀
u
∈
E
~
0
(
Ω
)
;
(4)
there exists a constantA
>
0 such that
(19)
∫
Ω
‍
-
χ
u
d
μ
≤
C
2
max
⁡
1
,
∫
0
∞
s
n
χ
′
(
-
s
)
c
a
p
Ω
(
u
<
-
s
)
d
s
1
/
n
,
1
,
∫
0
∞
s
n
χ
′
(
-
s
)
ca
p
Ω
(
u
<
-
s
)
d
s
1
/
n
kkii
∀
u
∈
E
0
(
Ω
)
;
(5)
there exists a locally bounded functionF
:
R
+
→
R
+ such that lim
⁡
⁡
sup
⁡
t
→
+
∞
F
(
t
)
/
t
<
1 and
(20)
∫
Ω
-
χ
(
u
)
d
μ
≤
F
C
χ
(
u
)
,
∀
u
∈
E
0
(
Ω
)
.Here E
~
0
(
Ω
) denotes the class E
~
0
(
Ω
)
∶
=
{
u
∈
E
0
(
Ω
)
;
∫
0
∞
s
n
χ
′
(
-
s
)
c
a
p
Ω
(
u
<
-
s
)
d
s
≤
1
} and C
χ
(
u
)
:
=
∫
0
∞
s
n
χ
′
(
-
s
)
c
a
p
Ω
(
u
<
-
s
)
d
s.The equivalences(
1
)
⇔
(
3
)
⇔
(
4
) are proved in [11] (Theorem 5.1) and the implication (
5
)
⇒
(
1
) is proved in [12] (Theorem 5.2). For the sake of completeness we include a complete proof.Proof.
We start by the implication(
1
)
⇒
(
2
). Let u
,
φ
∈
E
χ
(
Ω
). It follows from Proposition 5 that u
+
φ
∈
E
χ
(
W
). Hence
(21)
∫
Ω
-
χ
(
u
)
d
d
c
φ
n
≤
∫
Ω
-
χ
(
u
+
φ
)
d
d
c
(
u
+
φ
)
n
<
∞
.
Now, for the implication (
2
)
⇒
(
3
), assume that (3) is not satisfied. Then for each j
∈
N we can find a function u
j
∈
E
~
0
(
Ω
) such that
(22)
∫
Ω
‍
-
χ
(
u
j
)
d
μ
≥
2
3
j
.
Consider the function
(23)
u
:
=
∑
j
=
1
∞
1
2
2
j
u
j
.
Observe that
(24)
(
u
<
-
s
)
⊂
⋃
1
∞
u
j
<
-
2
j
s
.
Hence
(25)
ca
p
Ω
(
u
<
-
s
)
≤
∑
j
∞
ca
p
Ω
(
u
j
<
-
2
j
s
)
.
Now, since the weight χ is convex or homogeneous and using the estimates (14), we get
(26)
∫
0
∞
s
n
χ
′
(
-
s
)
ca
p
Ω
(
u
<
-
s
)
d
s
≤
∫
0
∞
s
n
χ
′
(
-
s
)
∑
j
∞
ca
p
Ω
(
u
j
<
-
2
j
s
)
d
s
≤
∑
j
∞
1
2
n
j
-
n
∫
0
∞
2
j
-
1
s
n
χ
′
(
-
s
)
ca
p
Ω
(
u
j
<
-
2
j
s
)
≤
∑
j
∞
1
2
n
j
-
n
∫
0
∞
2
j
-
1
s
n
χ
′
(
-
s
)
ca
p
Ω
(
u
j
<
-
s
)
≤
2
n
∑
j
∞
1
2
n
j
<
∞
.
Hence u
∈
E
χ
(
Ω
). On the other hand, from (22) we have
(27)
∫
Ω
-
χ
(
u
)
d
μ
≥
1
2
2
j
∫
Ω
-
χ
(
u
j
)
d
μ
≥
2
j
,
∀
j
∈
N
,
which yields a contradiction.
Now, we prove that(
3
)
⇒
(
4
). Let ψ
∈
E
0
(
Ω
), denote E
χ
(
ψ
)
:
=
∫
Ω
-
χ
(
ψ
)
(
d
d
c
ψ
)
n. If ψ
∈
E
~
0
(
Ω
), that is, C
χ
(
ψ
)
≤
1, then
(28)
∫
Ω
-
χ
(
ψ
)
d
μ
≤
2
n
=
C
.
If C
χ
(
ψ
)
>
1 the function ψ
~ defined by
(29)
ψ
~
∶
=
ψ
1
+
C
χ
ψ
1
/
n
∈
E
~
0
(
Ω
)
.
Indeed, from the monotonicity of χ, we have
(30)
∫
0
∞
χ
′
(
s
)
s
n
ca
p
Ω
ψ
1
+
C
χ
ψ
1
/
n
<
-
s
d
s
=
1
C
χ
ψ
∫
0
∞
χ
′
(
s
)
s
C
χ
ψ
1
/
n
n
1
C
χ
ψ
f
f
f
·
ca
p
Ω
ψ
<
-
s
-
s
C
χ
ψ
1
/
n
d
s
≤
1
C
χ
(
ψ
)
∫
0
∞
χ
′
(
s
)
s
n
ca
p
Ω
ψ
<
-
s
d
s
=
1
.
It follows from (18) and the convexity of χ that
(31)
∫
Ω
-
χ
(
ψ
)
d
μ
≤
2
C
χ
ψ
1
/
n
∫
Ω
-
χ
ψ
1
+
C
χ
ψ
1
/
n
d
μ
≤
A
C
χ
ψ
1
/
n
.
Hence we get (19).
For the implication(
4
)
⇒
(
5
), we consider F
(
t
)
=
A
max
⁡
(
1
,
t
1
/
n
).
(
5
)
⇒
(
1
). It follows from [12] (Theorem 4.5) that the class E
χ
(
Ω
) characterizes pluripolar sets in the sense that if P is a locally pluripolar subset of Ω then P
⊂
{
v
=
-
∞
}, for some v
∈
E
χ
(
Ω
). Then assumption (20) on μ implies that it vanishes on pluripolar sets. It follows from [13] that there exists a function u
∈
E
0
(
Ω
) and f
∈
L
loc
⁡
1
(
(
d
d
c
u
)
n
) such that μ
=
f
(
d
d
c
u
)
n.
Considerμ
j
∶
=
min
⁡
(
f
,
j
)
(
d
d
c
u
)
n. This is a finite measure which is bounded from above by the complex Monge-Ampère measure of a bounded function. It follows therefore from [3] that there exist φ
j
∈
E
0
(
Ω
) such that
(32)
d
d
c
φ
j
n
=
min
⁡
f
,
j
d
d
c
u
n
.
The comparison principle shows that φ
j is a decreasing sequence. Set φ
=
lim
⁡
j
→
∞
φ
j. It follows from (20) that
(33)
∫
Ω
-
χ
(
φ
j
)
d
d
c
φ
j
n
≤
F
∫
0
∞
χ
′
s
s
n
ca
p
Ω
φ
j
<
-
s
d
s
.
Hence
(34)
sup
⁡
j
∫
0
∞
χ
′
(
s
)
s
n
ca
p
Ω
φ
j
<
-
s
d
s
<
∞
.
This implies that
(35)
∫
0
+
∞
t
n
χ
′
(
-
t
)
ca
p
Ω
(
{
φ
<
-
t
}
)
d
t
<
+
∞
.
Then φ
≢
-
∞ and therefore φ
∈
E
χ
(
Ω
).
We conclude now by continuity of the complex Monge-Ampère operator along decreasing sequences that(
d
d
c
φ
)
n
=
μ. The uniqueness of φ follows from the comparison principle.


## 3. The Weighted Energy Class with Boundary Values

Letχ
:
R
-
→
R
- be a nondecreasing function and let f
∈
M
(
Ω
) be a maximal psh function. We define the class E
χ
(
f
) (resp., N
f, F
(
f
), N
a
(
f
), and F
a
(
f
)) to be the class of psh functions u such that there exists a function φ
∈
E
χ
(
Ω
) (resp., N
,
F
,
N
a
,
F
a) such that
(36)
φ
(
z
)
+
f
(
z
)
≤
u
(
z
)
≤
f
(
z
)
,
∀
z
∈
Ω
.
Later on, we will use repeatedly the following well known comparison principle from [1] as well as its generalizations to the class N
(
f
) (cf. [10, 14]).Theorem 7 (see [1, 10, 14]).
Letf
∈
E
(
Ω
) be a maximal function and u
,
v
∈
N
(
f
) be such that (
d
d
c
u
)
n vanishes on all pluripolar sets in Ω. Then
(37)
∫
(
u
<
v
)
d
d
c
v
n
≤
∫
(
u
<
v
)
d
d
c
u
n
.
Furthermore if (
d
d
c
u
)
n
=
(
d
d
c
v
)
n then u
=
v.The following lemma, which gives an estimate of the size of sublevel set in terms of the mass of Monge-Ampère measure, will be useful shortly.Lemma 8.
Letχ
:
R
-
→
R
- be a nondecreasing function such that χ
(
t
)
<
0, for all t
<
0, and f
∈
E a maximal function. Then for all φ
∈
E
χ
(
f
)
(38)
t
n
c
a
p
Ω
(
φ
<
-
s
-
t
+
f
)
≤
∫
φ
<
-
s
+
f
d
d
c
φ
n
,
∀
s
>
0
,
t
>
0
.Proof.
Fixs
,
t
>
0. Let K
⊂
{
φ
<
f
-
s
-
t
} be a compact subset. Then
(39)
ca
p
Ω
K
=
∫
Ω
d
d
c
u
K
*
n
=
∫
(
φ
<
f
-
s
-
t
)
d
d
c
u
K
*
n
=
∫
(
φ
<
f
-
s
+
t
u
K
*
)
d
d
c
u
K
*
n
≤
1
t
n
∫
{
φ
<
v
}
d
d
c
v
n
,
where u
K
* is the relative extremal function of the compact K and v
∶
=
f
-
s
+
t
u
K
*. It follows from Theorem 7 that
(40)
1
t
n
∫
{
φ
<
v
}
d
d
c
v
n
=
1
t
n
∫
{
φ
<
max
⁡
(
φ
,
v
)
}
d
d
c
max
⁡
(
φ
,
v
)
n
≤
1
t
n
∫
{
φ
<
max
⁡
(
φ
,
v
)
}
d
d
c
φ
n
=
1
t
n
∫
{
φ
<
f
-
s
+
t
u
K
}
d
d
c
φ
)
n
≤
1
t
n
∫
{
φ
<
f
-
s
}
d
d
c
φ
)
n
.
Taking the supremum over all K’s yields the first inequality.Proposition 9.
Letχ
:
R
-
→
R
- be an increasing function. Then one has
(41)
E
χ
f
⊂
u
∈
P
S
H
(
Ω
)
;
u
≤
f
,
∫
0
+
∞
s
n
χ
′
(
-
s
)
Ca
p
Ω
(
u
<
f
-
2
s
)
d
s
f
f
i
f
∫
0
+
∞
s
n
χ
′
(
-
s
)
c
a
p
Ω
(
u
<
f
-
2
s
)
d
s
<
+
∞
.
In particular, if χ
≢
0, then c
a
p
Ω
(
u
<
f
-
s
)
<
+
∞ for all s
>
0 and u
∈
E
χ
(
f
).Proof.
Letu
∈
E
χ
(
f
). Then there exists a function φ
∈
E
χ
(
Ω
) such that φ
+
f
≤
u. Therefore (
u
<
f
-
s
)
⊂
(
φ
<
-
s
). It follows from Lemma 8
(42)
∫
0
+
∞
s
n
χ
′
(
-
s
)
ca
p
Ω
(
u
<
f
-
2
s
)
d
s
≤
∫
0
+
∞
s
n
χ
′
(
-
s
)
ca
p
Ω
(
φ
<
-
2
s
)
d
s
≤
∫
0
+
∞
χ
′
(
-
s
)
∫
(
φ
<
-
s
)
d
d
c
φ
n
d
s
=
∫
Ω
-
χ
(
φ
)
d
d
c
φ
n
<
∞
.Theorem 10.
Letχ
:
R
-
→
R
- be an increasing function which satisfies χ
(
-
∞
)
=
-
∞ and f
∈
E a maximal function. Then if there exists a decreasing sequence u
j
∈
E
0
(
f
) such that
(43)
sup
⁡
j
∫
Ω
-
χ
(
u
j
-
f
)
d
d
c
u
j
n
<
∞
then u
:
=
lim
⁡
j
→
∞
u
j
∈
E
χ
(
f
) and χ
(
u
-
f
)
∈
L
1
(
(
d
d
c
u
)
n
).
Conversely, ifu
∈
E
χ
(
f
) and χ
(
u
-
f
)
∈
L
1
(
(
d
d
c
u
)
n
) then there exists sequence u
j
∈
E
0
(
f
) decreasing towards u such that
(44)
sup
⁡
j
∫
Ω
-
χ
(
u
j
-
f
)
d
d
c
u
j
n
<
∞
.Proof.
Assume that the sequenceu
j
∈
E
0
(
f
)
∩
C
(
Ω
) (if necessary, we approximate u
j by a continuous sequence u
j
k
∈
E
0
(
f
)). For a fixed j
∈
N, let φ
j denote the function defined by
(45)
φ
j
(
z
)
:
=
sup
⁡
v
∈
PSH
(
Ω
)
;
v
+
f
≤
u
j
,
∀
z
∈
Ω
.
We claim that(1)
φ
j
∈
E
0;
(2)
(
d
d
c
φ
j
)
n
≤
(
d
d
c
u
j
)
n;
(3)
(
d
d
c
φ
j
)
n
=
0 on the subset (
φ
j
+
f
<
u
j
).Then, it follows from statements (1), (2), and (3) that for each j
∈
N
(46)
∫
Ω
-
χ
φ
j
n
d
d
c
φ
j
n
=
∫
(
φ
j
+
f
=
u
j
)
-
χ
φ
j
n
d
d
c
φ
j
n
≤
∫
Ω
-
χ
u
j
-
f
n
d
d
c
u
j
n
.
Therefore
(47)
sup
⁡
j
∫
Ω
-
χ
φ
j
n
d
d
c
φ
j
n
≤
sup
⁡
j
∫
Ω
-
χ
u
j
-
f
n
d
d
c
u
j
n
<
+
∞
.
Hence, the function φ
:
=
lim
⁡
j
→
∞
φ
j satisfies φ
+
f
≤
u
≤
f.
For the converse implication, fixu
∈
E
χ
(
f
). Then there exists a function φ
∈
E
χ such that φ
+
f
≤
u
≤
f. Let φ
j
∈
E
0
∩
C
(
Ω
) be a decreasing sequence with limit function φ. Then for each j
∈
N, consider the function u
j
∶
=
max
⁡
(
φ
j
+
f
,
u
)
∈
E
χ
(
f
). The sequence u
j decreases towards u and
(48)
∫
Ω
-
χ
u
j
-
f
n
d
d
c
u
j
n
≤
C
∫
Ω
-
χ
u
-
f
n
d
d
c
u
n
<
∞
,
where C is a constant which depends only on u and the proof of the theorem is completed.Theorem 11.
Letμ be a nonnegative measure in Ω, let χ
:
R
-
→
R
- be an increasing convex (or homogeneous) function such that χ
(
-
∞
)
=
-
∞, and let f
∈
E
(
Ω
) be a maximal function. Then there exists a unique function φ
∈
E
χ
(
f
) such that μ
=
(
d
d
c
φ
)
n if and only if μ satisfies one of the conditions of Theorem 6.Proof.
Assume thatμ
=
(
d
d
c
v
)
n for some v
∈
E
χ. Let (
Ω
j
)
j be a fundamental sequence of strictly pseudoconvex subsets of Ω. Choose a sequence f
j
∈
PSH
(
Ω
)
∩
C
(
Ω
-
) decreasing towards f on Ω and f
j is maximal on Ω
j
+
1. It follows from [13] that there exist a function g
∈
E
0 and a function θ
∈
L
loc
⁡
1
(
d
d
c
g
)
n such that
(49)
μ
=
θ
d
d
c
g
n
.
Consider the measure μ
j
=
⊩
Ω
j
min
⁡
(
θ
,
j
)
(
d
d
c
g
)
n, where ⊩
Ω
j denotes the characteristic function of the set Ω
j. Now, solving the Dirichlet problem in the strictly pseudoconvex domain Ω
j, we state that there exist functions u
j
,
v
j
∈
PSH
(
Ω
j
)
∩
C
(
Ω
-
j
) such that
(50)
d
d
c
u
j
n
=
d
d
c
v
j
n
=
μ
j
,
v
j
=
0
,
u
j
=
f
j
fggggggi
on
⁡
∂
Ω
j
.
By the comparison principle, we have u
j and v
j are decreasing sequences and
(51)
v
+
f
≤
v
j
+
f
j
≤
u
j
≤
f
j
on
⁡
Ω
j
.
Letting j
→
+
∞ we get that u
∶
=
lim
⁡
j
→
∞
u
j
∈
E
χ
(
f
). The continuity of the complex Monge-Ampère operator under monotonic sequences yields that (
d
d
c
u
)
n
=
μ. Uniqueness of u follows from the comparison principle.Corollary 12.
Letμ be nonnegative measure in Ω with total finite mass μ
(
μ
)
<
∞ and let f be a maximal function. Then there exists a uniquely determined function φ
∈
F
a
(
f
) such that (
d
d
c
φ
)
n
=
μ if and only if μ vanishes on pluripolar subsets.Proof.
It follows from [13] that there exist a function ψ
∈
E
0 and a function θ
∈
L
loc
⁡
1
(
d
d
c
ψ
)
n such that
(52)
μ
=
θ
d
d
c
ψ
n
.
By [3], there exists a unique h
j
∈
E
0 such that (
d
d
c
h
j
)
n
=
min
⁡
(
θ
,
j
)
(
d
d
c
ψ
)
n. The comparison principle yields that h
j is a decreasing sequence. Let denote by h
∶
=
lim
⁡
j
→
∞
h
j. It follows from Lemma 8 that h
≢
-
∞. Therefore h
∈
F
a. By the continuity of the complex Monge-Ampère operator under decreasing sequences, we have (
d
d
c
h
)
n
=
μ. Now, since
(53)
F
a
=
⋃
χ
convex
;
χ
(
0
)
≠
0
χ
(
-
∞
)
=
-
∞
E
χ
,
then there exists a convex function χ
:
R
-
→
R
- with χ
(
0
)
≠
0 and χ
(
-
∞
)
=
-
∞ such that h
∈
E
χ. By Theorem 11, we can find a unique function φ
∈
E
χ
(
f
)
⊂
F
a
(
f
) such that (
d
d
c
φ
)
n
=
μ.


## 4. Measures Dominated by Capacity

Throughout this section,μ denotes a fixed nonnegative measure of finite total mass μ
(
Ω
)
<
+
∞. We want to solve the Dirichlet problem
(54)
d
d
c
φ
n
=
μ
,
with
φ
∈
F
a
f
,
φ
|
∂
Ω
=
f
,
and measure how far the distance between the solution φ and the given boundary data f is from being bounded, by assuming that μ is suitable dominated by the Monge-Ampère capacity.Measures dominated by the Monge-Ampère capacity have been extensively studied by Kołodziej in [3–5]. The main result of his study, achieved in [4], can be formulated as follows. Fix ε
:
R
→
[
0
,
+
∞
[ a continuous decreasing function and set F
ε
(
x
)
∶
=
x
[
ε
(
-
ln
⁡
x
/
n
)
]
n. If for all compact subsets K
⊂
Ω
(55)
μ
K
≤
F
ε
ca
p
Ω
K
,
where
∫
0
+
∞
ε
(
t
)
d
t
<
+
∞
,
and l
:
∂
Ω
→
R is a continuous function, then μ
=
(
d
d
c
φ
)
n for somecontinuous function φ
∈
PSH
(
Ω
) with φ
|
∂
Ω
=
l.The condition∫
0
+
∞
ε
(
t
)
d
t
<
+
∞ means that ε decreases fast enough towards zero at infinity. This gives a quantitative estimate on how fast ε
(
-
lnCa
p
Ω
(
K
)
/
n
), hence μ
(
K
) decreases towards zero as ca
p
Ω
(
K
)
→
0.When∫
0
+
∞
ε
(
t
)
d
t
=
+
∞, it is still possible to show that μ
=
(
d
d
c
φ
)
n for some function φ
∈
F
(
Ω
), but φ will generally be unbounded. We now measure how far it is from being so.Theorem 13.
Letμ be a nonnegative finite measure. Assume for all compact subsets K
⊂
Ω
(56)
μ
(
K
)
≤
F
ε
c
a
p
Ω
(
K
)
.
Then there exists a unique function φ
∈
F
a
(
f
) such that μ
=
(
d
d
c
φ
)
n, and
(57)
c
a
p
Ω
φ
<
f
-
s
≤
exp
⁡
-
n
H
-
1
s
,
∀
s
>
0
.
Here H
-
1 is the reciprocal function of H
(
x
)
=
e
∫
0
x
ε
(
t
)
d
t
+
e
ε
(
0
)
+
μ
(
Ω
)
1
/
n.The proof is almost the same as that of Theorem 5.1 in [12], except that we use Corollary 12 for the existence of the solution and Lemma 8 to estimate the capacity of sublevel set.Observe that if∫
∞
ε
(
t
)
d
t
<
∞ then H is bounded by e
∫
0
∞
ε
(
t
)
d
t
+
e
ε
(
0
)
+
μ
(
Ω
)
1
/
n. Hence H
-
∞
(
t
)
=
+
∞, ∀
t
≥
e
∫
0
∞
ε
(
t
)
d
t
+
e
ε
(
0
)
+
μ
(
Ω
)
1
/
n. Therefore
(58)
0
≤
f
-
φ
≤
e
∫
0
∞
ε
(
t
)
d
t
+
e
ε
(
0
)
+
μ
Ω
1
/
n
.Now, we consider the case whenμ
=
f
d
λ is absolutely continuous with respect to Lebesgue measure.LetG
⊂
C
n denote a generic subspace of C
n that is a real subspace such that G
+
J
G
=
C
n, where J is the usual complex structure on C
n (cf. [21] for more details). G will be endowed with the induced Euclidean structure and the corresponding Lebesgue measure which will be denoted by λ
G.Letα
>
0 be a positive real number. According to [22, 23], the Orlicz space L
lo
g
n
+
α
L
(
d
λ
G
) consists of λ
G-measurable functions g defined on Ω
∩
G such that
(59)
∫
Ω
∩
Γ
f
λ
log
⁡
n
+
α
1
+
f
λ
d
λ
G
<
∞
,
for
some
λ
>
0
.
On the space L
lo
g
n
+
α
L
(
d
μ
), we define the norm
(60)
f
L
lo
g
n
+
α
L
∶
=
inf
⁡
λ
>
0
;
∫
Ω
|
f
|
λ
log
⁡
n
+
α
(
e
+
|
f
|
λ
)
d
λ
G
<
1
.
The dual space to L
lo
g
n
+
α
L is the exponential class Exp
L
1
/
n
+
α; that is, the vector space
(61)
Exp
L
1
/
n
+
α
∶
=
f
:
Ω
⟶
R
¯
;
∃
λ
>
0
:
∫
Ω
exp
⁡
|
f
|
λ
1
/
(
n
+
α
)
-
1
d
λ
G
<
∞
b
g
g
g
∫
Ω
exp
⁡
|
f
|
λ
1
/
n
+
α
-
1
d
λ
G
<
∞
equipped with the norm
(62)
f
Exp
L
1
/
n
+
α
∶
=
inf
⁡
λ
>
0
;

∫
Ω
exp
⁡
|
f
|
λ
1
/
n
+
α
-
1
d
λ
G

<

1
.
Then we have the following Hölder inequality:
(63)
∫
Ω
‍
f
g
d
λ
G
≤
C
n
,
α
f
L
lo
g
n
+
α
L
g
Exp
L
1
/
n
+
α
,
for f
∈
L
lo
g
n
+
α
L and g
∈
Exp
L
1
/
n
+
α, where C
n
,
α
>
0 is a positive constant depending only on n and α. By a simple computation, we have
(64)
1
Exp
L
1
/
n
+
α
(
K
)
=
1
log
⁡
n
+
α
1
+
1
/
λ
G
(
K
)
.Corollary 14.
Letμ
=
⊩
Ω
∩
G
g
λ
G be a measure with nonnegative density g
∈
L
log
⁡
n
+
α
L
(
Ω
∩
G
). Then there exists a unique bounded function φ
∈
F
a
f
∩
L
∞
Ω such that (
d
d
c
φ
)
n
=
μ and
(65)
0
≤
f
-
φ
≤
C
g
L
log
⁡
n
+
α
L
1
/
n
,
where C
>
0 only depends on n
,
α
,
Ω, and G.Proof.
We claim that there exists a constantC
>
0 such that
(66)
μ
K
≤
C
g
L
log
⁡
n
+
α
L
1
/
n
n
ca
p
Ω
(
α
+
n
)
/
n
K
,
ggggggrrribgbg
∀
compact
K
⊂
Ω
.
Indeed, Hölder’s inequality and inequality (66) yield
(67)
μ
K
≤
g
L
log
⁡
n
+
α
L
1
log
⁡
n
+
α
1
+
1
/
λ
G
K
,
rrfiiiiiiiiiiiiirrruuy
∀
compact
K
⊂
Ω
.
By [21] we have
(68)
λ
G
K
≤
C
exp
⁡
-
1
ca
p
Ω
1
/
n
K
,
∀
compact
K
⊂
Ω
,
where C
>
0 is a constant which depends only on Ω and G.
Inequality (66) follows by combining (67) and (68).
Then we apply Theorem13 with
(69)
ε
(
x
)
=
C
g
L
log
⁡
n
+
α
L
C
g
L
log
⁡
n
+
α
L
1
/
n
e
-
α
x
/
n
,
which yields
(70)
0
≤
f
-
φ
≤
e
∫
0
x
ε
(
t
)
d
t
+
e
ε
(
0
)
+
μ
Ω
1
/
n
≤
C
g
L
log
⁡
n
+
α
L
1
/
n
.



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*Source: 947819-2015-02-10.xml*