Tóm tắt Luận án Một số vấn đề trong lý thuyết đa thế vị

 negative answer in the case of subspaces of higher co-dimension.
Example 1.4.4 (Counterexample). In Rn (n ≥ 3), we denote by Bn−2(0, R) the
open ball in Rn−2, with center at 0 and radius R > 0. For i ≥ 2, let µi be the
measure defined on
(
Bn−2(0, i)Bn−2(0, 1i )
)×{0}×{0} generated by the (n−2)-
dimensional Lebesgue measure on
(
Bn−2(0, i)Bn−2(0, 1i )
)
. We define potentials
pµi : Rn −→ [−∞,∞) by
pµi(x) =
∫
Rn
−1
|x− w|n−2dµi(w).
Then we obtain the sequence {pµi}i≥2 ⊂ SH(Rn) which satisfies properties:
pµi ≤ 0 on Rn,
pµi = −∞ on
(
Bn−2(0, i)Bn−2(0, 1i )
)× {0} × {0},
−∞ < pµi(0) < 0,
for all i ≥ 2. By setting
ui(x) = −pµi(x)
pµi(0)
,
the sequence {ui}i≥2 ⊂ SH(Rn) has properties:
ui ≤ 0 on Rn,
ui = −∞ on
(
Bn−2(0, i)Bn−2(0, 1i )
)× {0} × {0},
ui(0) = −1,
for all i ≥ 2. Now we define
u(x) =
∞∑
i=2
1
2i−1
ui(x),
hence 
u ∈ SH(Rn),
u = −∞ on (Rn−2{0})× {0} × {0},
u(0) = −1.
15
Therefore, the function u˜ = max(u,−2) satisfies:
u˜ ∈ SH(Rn),
u˜ = −2 on (Rn−2{0})× {0} × {0},
u˜(0) = −1.
Finally, by setting v˜ ≡ −2, we conclude that v˜ ≥ u˜ almost everywhere on Rn−2×
{0} × {0} with respect to (n − 2)-dimensional the Lebesgue measure on Rn−2 ×
{0} × {0}, but not everywhere as v˜(0) < u˜(0).
1.5 Other versions of main results
This section presents the main results (Theorem 1.3.3, Theorem 1.4.1) in term
of measure densities. Given a Borel measure η on Rn and a Borel setK, we denote
ηK(E) = η(K ∩ E)
for convenience. It is clear that ηK is also a Borel measure on Rn.
The main idea in the proof of Theorem 1.3.3 is to bound the functions fϵ by a
function that is integrable with respect to measure ν. Now, with some conditions
on upper and lower densities of measure µ, we obtain another version of Theorem
1.3.3 as follows.
Theorem 1.5.1. Let Ω be a domain in Rn (n ≥ 2) and u ∈ SH(Ω). Let K be
a Borel subset of Ω and x0 be a point of K. Suppose that there exist a positive
Borel measure µ, a relatively compact open subset U of Ω that contains x0 and a
positive number s > n− 2 satisfying
Θs∗(µK, x0) > 0 and
∫
U
Θ∗s(µK, w)
|w|n−2 dν(w) < +∞
where ν = ∆u|U . Then
lim
ϵ→0
1
µ
(
K ∩ B(x0, ϵ)
) ∫
K∩B(x0,ϵ)
u(x)dµ(x) = u(x0).
Proof. We can assume that x0 = 0 and U = B(0, 1). Following the proof technique
of Theorem 1.3.3, it is sufficient to show that there exist constants C1, C2, C3
satisfying
fϵ(w) ≤ C1g(|w|) + C2
Θs∗(µK, 0)
.
Θ∗s(µK, w)
|w|n−2 + C3,
16
for all w ∈ B(0, 1){0} and ϵ > 0 small enough. For p > 1, we also consider two
cases as before. The proof differs only in the case II where |w| ≤ pϵ. In this case,
we have
fϵ(w) ≤ 1
µ(Kϵ)
α(ϵ)∫
0
µ(Kϵ)dt+
1
µ(Kϵ)
+∞∫
α(ϵ)
µ(K ∩ B(w, 2g−1(t)))dt
≤ α(ϵ) + 1
(2ϵ)sΘs∗(µK, 0)
+∞∫
α(ϵ)
(4g−1(t))sΘ∗s(µK, w)dt
≤ g(|w|) + 2
smax{1, n− 2}
s− (n− 2) .
Θ∗s(µK, w)
Θs∗(µK, 0)
.
( |w|+ ϵ
ϵ
)s
.
1
(|w|+ ϵ)n−2
≤ g(|w|) + 2
s.(p+ 1)smax{1, n− 2}
s− (n− 2) .
Θ∗s(µK, w)
Θs∗(µK, 0)
.
1
|w|n−2 ,
as desired.
The next result, as a consequence of Theorem 1.5.1, is another version of The-
orem 1.4.1. Their proofs are the same.
Theorem 1.5.2. Let Ω be a domain in Rn (n ≥ 2) and K be a Borel subset of
Ω. Let u be an upper semicontinuous function and v be a subharmonic function
in Ω. Suppose that there exist a positive Borel measure µ and a positive number
s > n − 2 such that for all x ∈ K, there exists a relatively compact open subset
Ux of Ω that contains x satisfying:
Θs∗(µK, x) > 0 and
∫
Ux
Θ∗s(µK, w)
|w|n−2 dν(w) < +∞
where ν = ∆v|Ux. If u ≥ v almost everywhere on K with respect to µ then u ≥ v
on K.
17
Chapter 2
Complex Monge-Ampère equation in
strictly pseudoconvex domains
Written on the basis of the paper “Hoang Son Do, Thai Duong Do, Hoang Hiep
Pham, Complex Monge-Ampère equation in strictly pseudoconvex domains, Acta
Math. Vietnam. 45 (2020), 93–101”, the present chapter is devoted to study
the Dirichlet problem for the complex Monge-Ampère equation (ddcu)n = µ in
a strictly pseudoconvex domain Ω with the boundary condition u = ϕ, where
ϕ ∈ C(∂Ω). This chapter is organized as follows.
- In Section 2.1, Section 2.2 and Section 2.3, we recall some basic properties of
plurisubharmonic functions, some basic properties of relative capacity, domain
of Monge-Ampère operator and notions of Cegrell classes which will be used
in the sequel.
- In Section 2.4, we recall comparison principles and some sufficient conditions
for Dirichlet problem. After that, we prove our main result which uses these
results as main tools. This main result is also our attempt to study Problem
2 mentioned in the Introduction.
- In Section 2.5, we prove our results on studying Problem 3.
Throughout Chapter 2 and Chapter 3, we always assume that Ω is a domain in
Cn.
18
2.1 Some properties of plurisubharmonic functions
In this section, we recall the definition and basic properties of plurisubharmonic
functions, definition of hyperconvex domains and strictly pseudoconvex domains.
For more details, the reader is referred to [25, 32, 22].
Definition 2.1.1. Let u : Ω → [−∞,∞) be an upper semicontinuous function
which is not identically −∞. We say that u is plurisubharmonic if for each complex
line
{z1 + λz2 ∈ Ω : λ ∈ C},
the function λ 7→ u(z1 + λz2) is subharmonic or identically −∞ on the set {λ ∈
C : z1+λz2 ∈ Ω}. We denote by PSH(Ω) and PSH−(Ω) respectively the family
of all plurisubharmonic functions and the family of all negative plurisubharmonic
functions in Ω.
Similarly as in the case of subharmonic functions, we shall recall the main
approximation theorem for plurisubharmonic functions. The notation we use here
is the same as in Section 1.1.
Theorem 2.1.2. Let Ω be a domain in Cn and let u ∈ PSH(Ω). If ϵ > 0 such
that Ωϵ ̸= ∅, then u ∗ ρϵ ∈ C∞ ∩ PSH(Ωϵ). Moreover, u ∗ ρϵ monotonically
decreases with decreasing ϵ and
lim
ϵ→0
u ∗ ρϵ(z) = u(z)
for each z ∈ Ω.
The following properties of families of plurisubharmonic functions is an imme-
diate consequence of its definition.
Theorem 2.1.3. Let Ω be a domain in Cn.
1, The family PSH(Ω) is a convex cone, i.e. if u, v ∈ PSH(Ω) then αu+βv ∈
PSH(Ω) for all α, β ≥ 0.
2, If {uj}i∈N ⊂ PSH(Ω) is a decreasing sequence then u = lim
j→∞
uj ∈ PSH(Ω)
or u ≡ −∞.
3, Let {uα}α∈A ⊂ PSH(Ω) be such that its upper envelope u = supα∈A uα is
locally bounded above. Then the upper semicontinuous regularization u∗ is
plurisubharmonic in Ω and u = u∗ almost everywhere. Moreover, if ϵ > 0
such that Ωϵ ̸= ∅, then u∗ρϵ ∈ C∞∩PSH(Ωϵ), u∗ρϵ monotonically decreases
with decreasing ϵ and lim
ϵ→0
u ∗ ρϵ(z) = u∗(z) for each z ∈ Ω.
19
Plurisubharmonicity can also be characterized in terms of distributional deriva-
tives.
Theorem 2.1.4. Let Ω be a domain in Cn and u ∈ PSH(Ω). Then for each
ζ = (ζ1, ζ2, ..., ζn) ∈ Cn, we have
n∑
j,k=1
∂2u
∂zj∂zk
ζjζk ≥ 0
in Ω, in the sense of distributions, i.e.∫
Ω
u(z)
n∑
j,k=1
∂2ϕ
∂zj∂zk
ζjζkdV2n(z) ≥ 0,
for any non-negative test functions ϕ ∈ C∞0 (Ω). Conversely, if v ∈ L1loc(Ω) is
such that for each ζ = (ζ1, ζ2, ..., ζn) ∈ Cn, we have
n∑
j,k=1
∂2v
∂zj∂zk
ζjζk ≥ 0
in Ω, in the sense of distributions, then the function u = lim
ϵ→0
(v∗ρϵ) is well-defined,
plurisubharmonic in Ω and equal to v almost everywhere in Ω.
By the main approximation theorem and characterizations of subharmonicity,
plurisubharmonicity, we have the following inclusion.
Corollary 2.1.5. If Ω is a domain in Cn, then PSH(Ω) ⊂ SH(Ω).
Similarly as in the case of subharmonic functions, we can sometimes glue plurisub-
harmonic functions together to give new plurisubharmonic functions.
Proposition 2.1.6. Let Ω be a domain in Cn, ω be a non-empty proper open
subset of Ω, and let u ∈ PSH(Ω), v ∈ PSH(ω). Suppose that
lim sup
x→y
v(y) ≤ u(y),
for each y ∈ ∂ω ∩ Ω. Then the formula
w =
max{u, v} in ω,u in Ω\ω
defines a plurisubharmonic function in Ω.
Plurisubharmonicity is preserved by holomorphic substitutions.
20
Proposition 2.1.7. Let Ω and Ω′ be domains in Cn and Cm, respectively. If
u ∈ PSH(Ω′) and f : Ω → Ω′ is a holomorphic mapping, then u ◦ f ≡ −∞ or
u ◦ f ∈ PSH(Ω).
We end this section with the definition of strictly plurisubharmonic functions,
maximal plurisubharmonic functions and pluripolar sets which will be used in the
next section.
Definition 2.1.8. Let Ω be a domain in Cn. A real-valued function u ∈ C2(Ω) is
said to be strictly plurisubharmonic if for every z ∈ Ω and 0 ̸= ζ ∈ Cn, we have
n∑
j,k=1
∂2u
∂zj∂zk
(z)ζjζj > 0.
Definition 2.1.9. Let Ω be a domain in Cn and let u : Ω → R be a plurisub-
harmonic function. We say that u is maximal if for every relatively compact open
subset U of Ω and every function ϕ ∈ PSH(U) ∩ USC(U¯), the following impli-
cation is true:
ϕ ≤ u on ∂U =⇒ ϕ ≤ u on U .
We denote by MPSH(Ω) the family of all maximal plurisubharmonic functions
in Ω.
Definition 2.1.10. A set E ⊂ Cn is called pluripolar if for each z ∈ E, there is
an open set U ∋ z and u ∈ PSH(U) such that E ∩ U ⊂ {u = −∞}.
2.2 Domain of Monge-Ampère operator and notions of
Cegrell classes
In this section, we recall the definition and some important properties of Monge-
Ampère operator. We also recall the definition of Cegrell’s classes and their gen-
eralizations which help us study the most general definition of the Monge-Ampère
operator and Dirichlet problem. For more details, we refer readers to the following
papers [1, 5, 9, 10, 12, 13, 14, 18, 24, 28, 34].
On Cn, we write d = ∂ + ∂ and dc = i(∂ − ∂), so that
ddcu = 2i∂∂u.
21
We denote by
dV =
( i
2
)n n∏
j=1
dzj ∧ dzj
the usual volume form. In the classical sense, if u is a smooth plurisubharmonic
function on a domain of Cn then its Monge-Ampère operator is a regular Radon
measure given by
(ddcu)n = cn det
( ∂2u
∂zj∂zj
)
dV,
where cn > 0 depends only on n and the power on the left is taken with respect to
the wedge product. Shiffman and Taylor have found an example showing that we
can not define (ddcu)n as a regular Radon measure for arbitrary plurisubharmonic
function u if n ≥ 2. Kiselman has simplified this example as follows.
Example 2.2.1. In Cn, we define
u(z) = (− log |z1|)1/n(|z2|2 + ...+ |zn|n − 1).
Then u is plurisubharmonic near the origin, smooth away from the hyperplane
z1 = 0, but (ddcu)n is unbounded near z1 = 0.
On the other hand, Bedford and Taylor have defined (ddcu)n as a closed positive
current in the case where u is a continuous plurisubharmonic function and then in
the case where u is a locally bounded plurisubharmonic function. In general, the
operator
(u1, ..., un) 7→ ddcu1 ∧ ... ∧ ddcun,
which is also referred to as the Monge-Ampère operator, is well-defined if u1, ..., un
are locally bounded plurisubharmonic functions on a domain of Cn. We recall some
important properties of the Monge-Ampère operator acting on locally bounded
plurisubharmonic functions. The first result is known as Chern-Levine-Nirenberg
inequality.
Theorem 2.2.2. Let Ω and Ω′ be domains in Cn such that Ω′ b Ω. Then there
exists a constant C > 0 which depends only on Ω and Ω′ such that∫
Ω′
ddcu1 ∧ ... ∧ ddcun ≤ C‖u1‖Ω...‖un‖Ω,
for any set {uj}nj=1 ⊂ PSH(Ω) ∩ L∞loc(Ω), where ‖ · ‖ denotes the sup norm of a
function.
22
The second result, known as convergence theorem, shows that monotone con-
vergence of locally bounded plurisubharmonic functions implies the convergence
of the corresponding currents.
Theorem 2.2.3. Let Ω be a domain in Cn and u1, u2, ..., un ∈ PSH(Ω)∩L∞loc(Ω).
Let {ujk}j∈N ⊂ PSH(Ω) be an increasing (or a decreasing) sequence for k =
1, 2, ..., n such that lim
j→∞
ujk = uk almost everywhere, for every k. Then
ddcuj1 ∧ ... ∧ ddcujn → ddcu1 ∧ ... ∧ ddcun
in the weak topology of currents.
After that, Demailly has proved that the domain of Monge-Amère operator can
be extended to the set of all plurisubharmonic functions which are bounded near
the boundary of the domain. He also proved an improved version of the Chern-
Levine-Nirenberg inequality and a result on the continuity of the Monge-Ampère
operator.
In fact, since a plurisubharmonic function can be approximated by a decreasing
sequence of smooth plurisubharmonic functions (by Theorem 2.1.2), the continuity
of the Monge-Ampère operator under decreasing limits is a matter of interest. The
choice of decreasing sequences for considering continuity of the Monge-Ampère
operator is also motivated by the following example where Cegrell has constructed
two sequences of smooth plurisubharmonic functions which both decrease to a
plurisubharmonic function but the limits of their Monge-Ampère operators are
not the same.
Example 2.2.4. In Cn, let
u(z) = 2 log |z1...zn|,
uj(z) = log(|z1...zn|2 + 1/j),
vj(z) = log(|z1|2 + 1/j) + ...+ log(|zn|2 + 1/j).
We have  {uj}j∈N, {vj}j∈N ⊂ PSH ∩ C∞(Cn),uj ↘ u, vj ↘ u,
but (ddcuj)n and (ddcvj)n tend respectively weakly to 0 and n!4nδ0, where δ0 is the
Dirac measure at the origin.
Therefore, for an open subset Ω ⊂ Cn, it is natural to define the subclass
D(Ω) ⊂ PSH(Ω), for which the Monge-Ampère operator can be well-defined,
23
as follows: a plurisubharmonic function u belongs to D(Ω) if there exists a non-
negative Radon measure µ on Ω such that if Ω′ ⊂ Ω is open and a sequence
{uj}j∈N ⊂ PSH ∩ C∞(Ω′) decreases to u in Ω′ then (ddcuj)n tends weakly to µ
in Ω′. B locki has given a precise characterization of D(Ω) and has shown that the
Monge-Ampère operator is continuous for decreasing sequences in D(Ω).
On the other hand, with the development of notions of convexity, the Monge-
Ampère operator has been studied in hyperconvex domains and strictly pseudo-
convex domains which give more information than usual domains.
Definition 2.2.5. Let Ω be a domain in Cn. Then Ω is said to be a hyperconvex
domain if it admits an exhaustion function which is non-positive and plurisubhar-
monic, i.e. there exists ϕ ∈ PSH−(Ω) such that {z ∈ Ω : ϕ(z) < −c} b Ω for
all c > 0.
Definition 2.2.6. Let Ω be a bounded domain in Cn with C2−smooth boundary.
Then Ω is said to be strictly pseudoconvex if there exists a C2−smooth defining
function for Ω which is strictly plurisubharmonic on a neighbourhood of Ω.
Example 2.2.7. The set {(z, w) ∈ C2 : ‖z‖2 + ‖w‖4 < 1} is a hyperconvex
domain, but not a strictly pseudoconvex domain.
For a bounded hyperconvex domain, Cegrell introduced finite energy classes of
plurisubharmonic functions which are now known as Cegrell’s classes.
Definition 2.2.8. Let Ω be a bounded hyperconvex domain in Cn. Define
E0(Ω) = {u ∈ PSH−(Ω) ∩ L∞(Ω) : lim
z→∂Ω
u(z) = 0,
∫
Ω
(ddcu)n <∞},
F(Ω) = {u ∈ PSH−(Ω) : ∃ {uj} ⊂ E0(Ω), uj ↘ u, sup
j
∫
Ω
(ddcuj)
n <∞},
E(Ω) = {u ∈ PSH−(Ω) : ∀K b Ω,∃uK ∈ F(Ω) such that uK ≥ u on Ω
and uK = u on K},
N (Ω) = {u ∈ E(Ω) : the smallest maximal plurisubharmonic majorant = 0}.
It is clearly that E0 ⊂ F ⊂ N ⊂ E .
In the case where Ω is bounded hyperconvex domain, Cegrell has shown that
the class E(Ω) is the largest subclass of PSH−(Ω) on which the Monge-Ampère
operator is well-defined, in other words,
E(Ω) = D(Ω) ∩ PSH−(Ω).
We end this section with the following generalizations of the classes E0,F ,N
which have been used to study the Dirichlet problem with smooth boundary data.
24
Definition 2.2.9. Let Ω be a bounded strictly pseudoconvex domain in Cn.
i) Let H be a maximal plurisubharmonic function in Ω. For K ∈ {E0,F ,N}, we
denote
K(H) = K(H,Ω) = {u ∈ PSH(Ω) : ∃φ ∈ K, H ≥ u ≥ φ+H}.
ii) Let f ∈ C(∂Ω) such that
lim
z→ξ
U(0, f)(z) = f(ξ) for every ξ ∈ ∂Ω,
where
U(0, f) = sup{v ∈ PSH ∩ L∞loc(Ω) : lim sup
z→ξ
v(z) ≤ f(x) ∀ξ ∈ ∂Ω}.
For K ∈ {E0,F ,N}, we denote
K(f) = K(f,Ω) = {u ∈ PSH(Ω) : ∃φ ∈ K, U(0, f) ≥ u ≥ φ+ U(0, f)}.
2.3 Some basic properties of relative capacity
In this section, we recall the definition and some important properties of the
relative capacity, especially in strictly pseudoconvex domains. For more details,
the reader is referred to [3, 6, 25, 27, 35].
Definition 2.3.1. Let Ω be an open subset of Cn and K be a compact subset of
Ω. The relative capacity of K in Ω is defined by
Cap(K,Ω) = sup
{ ∫
K
(ddcv)n|v ∈ PSH(Ω, [−1, 0])
}
.
According to the Chern-Levine-Nirenberg inequality (Theorem 2.2.2), we have,
for every compact subset K ⊂ Ω,
Cap(K,Ω) <∞.
Definition 2.3.2. If E ⊂ Ω then the relative capacity of E in Ω is defined by
Cap(E,Ω) = sup{Cap(K,Ω)|K is a compact subset of E}.
We recall some properties of relative capacity.
Proposition 2.3.3. Let Ω,Ω′ be domains in Cn.
1, If E ⊂ Ω is a Borel set then
25
Cap(E,Ω) = sup
{ ∫
E
(ddcv)n|v ∈ PSH(Ω, [−1, 0])
}
.
2, If E ′ ⊂ E ⊂ Ω ⊂ Ω′ then
Cap(E,Ω) ≥ Cap(E ′,Ω) and Cap(E,Ω) ≥ Cap(E,Ω′).
3, If {Ej}j∈N ⊂ Ω then
Cap
( ∞⋃
j=1
Ej,Ω
)
≤
∞∑
j=1
Cap(Ej,Ω).
4, If E1 ⊂ E2 ⊂ ... are Borel subsets of Ω, then
Cap
( ∞⋃
j=1
Ej,Ω
)
= lim
j→∞
Cap(Ej,Ω).
5, If E is a pluripolar set then Cap(E,Ω) = 0.
Next, we recall some results of relative capacity when we require the assumption
of strictly pseudoconvexity.
Theorem 2.3.4. Let Ω′′,Ω′,Ω be bounded strictly pseudoconvex domains satisfy-
ing Ω′′ b Ω′ b Ω. Then, there exists a constant A > 0 such that
Cap(K,Ω) ≤ Cap(K,Ω′) ≤ A Cap(K,Ω),
for every compact subset K ⊂ Ω′′.
Theorem 2.3.5. Let Ω be a bounded strictly pseudoconvex domain and E be an
open subset of Ω. Then there exists a constant A > 0 which depends only on E
and Ω such that
V2n(E
′) ≤ ACap(E ′,Ω),
for every E ′ ⊂ E.
Relative capacity is used to describe the behaviour of plurisubharmonic func-
tions as follows.
Theorem 2.3.6. Let Ω be a bounded strictly pseudoconvex domain and u ∈
PSH(Ω). Then, for each ϵ > 0, there exists an open subset U ⊂ Ω with
Cap(U,Ω) < ϵ such that u|Ω\U is continuous.
By this result, we say that plurisubharmonic functions are quasicontinuous.
26
Theorem 2.3.7. Let Ω be a bounded strictly pseudoconvex domain and u ∈
PSH(Ω). If {uj}j∈N ⊂ PSH(Ω) is a decreasing sequence, convergent to u in Ω
and uj = u in a neighbourhood of ∂Ω then we have
lim
j→∞
Cap({uj > u+ ϵ},Ω) = 0,
for any ϵ > 0.
We end this section by the continuity of the Monge-Ampère operator on se-
quences converging with respect to capacity.
Theorem 2.3.8. Let Ω be a bounded strictly pseudoconvex domain and {uj}j∈N ⊂
PSH(Ω) be a uniformly bounded sequence converging with respect to capacity to
u ∈ PSH(Ω), i.e. for any ϵ > 0 and K b Ω,
lim
j→∞
Cap(K ∩ {|u− uj| > ϵ},Ω) = 0.
Then we have
(ddcuj)
n → (ddcu)n,
in the sense of currents.
2.4 Dirichlet problem for the Monge-Ampère equation
is strictly pseudoconvex
In this section, we establish a sufficient condition for the continuity of the
solution outside an analytic set. To prove this result, we need some auxiliary
results including some comparison principles and some sufficient conditions for
Dirichlet problem.
First, we recall the Bedford-Taylor comparison principle which reflects an "ellip-
tic" nature of the Monge-Ampère operator. The idea of this comparison principle
is to use the comparison between the Monge-Ampère operators of two plurisub-
harmonic functions u, v to compare u and v.
Theorem 2.4.1 (see [6]). Let Ω be a bounded open set in Cn. Let u, v ∈ PSH(Ω)∩
L∞(Ω) such that lim inf
Ω∋z→∂Ω
(u(z)− v(z)) ≥ 0. Then∫
{u<v}
(ddcv)n ≤
∫
{u<v}
(ddcu)n.
27
Theorem 2.4.1 has been generalized in several directions. One of improved
versions is the following.
Theorem 2.4.2 (see [2, 31]). Let Ω be a hyperconvex domain and u, v ∈ E(Ω).
Assume that one of the following conditions holds
(i) lim inf
Ω∋z→∂Ω
(u(z)− v(z)) ≥ 0.
(ii) u ∈ N (H,Ω) for some maximal plurisubharmonic function H ≤ 0, and
v ≤ H.
Then,
1
n!
∫
{u<v}
(v−u)nddcw1∧...∧ddcwn+
∫
{u<v}
−w1(ddcv)n ≤
∫
{u<v}∪{u=v=−∞}
−w1(ddcu)n,
for any w1, ..., wn ∈ PSH(Ω, [−1, 0]).
The f