the monge–ampère type equation in the weighted pluricomplex energy class
TRANSCRIPT
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International Journal of MathematicsVol. 25, No. 5 (2014) 1450042 (17 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0129167X14500426
The Monge–Ampere type equation in the weightedpluricomplex energy class
Le Mau Hai∗, Pham Hoang Hiep† and Nguyen Xuan Hong‡
Department of MathematicsHanoi National University of Education
Hanoi, Vietnam∗[email protected]
†phhiep [email protected]‡[email protected]
Nguyen Van Phu
Postgraduate DepartmentElectric Power University
Hanoi, [email protected]
Received 12 February 2013Accepted 19 March 2014Published 16 April 2014
In the paper, we prove the existence of solutions of the complex Monge–Ampere typeequation −χ(u)(ddcu)n = µ in the class Eχ(Ω) if there exist subsolutions in this class.As an application, we prove that the complex Monge–Ampere equation (ddcu)n = µ issolvable in the class E(Ω) if there exist subsolutions locally. Moreover, by an examplewe show that the conditions in our above result are sharp.
Keywords: Complex Monge–Ampere equation; plurisubharmonic functions; class Eχ;class N ; class E.
Mathematics Subject Classification 2010: 32U05, 32U15, 32W20
1. Introduction
Let Ω be a hyperconvex domain in Cn. As well-known, the complex Monge–Ampereoperator has a central role in pluripotential theory and has been extensively stud-ied for many years. In [2, 3], Bedford and Taylor has shown that this operatoris well defined on the class of locally bounded plurisubharmonic functions withrange in the class of non-negative measures. However, an example of Kiselman in[23] tells that it is not possible to extend this operator in a meaningful way tothe whole class of plurisubharmonic functions and still have the range containedin the class of non-negative measures. Therefore, the question of finding a naturaldomain of definition for the complex Monge–Ampere operator is very important
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for pluripotential theory. In [9] Cegrell defined the classes E0(Ω),Fp(Ω), Ep(Ω) onwhich the complex Monge–Ampere operator is well defined. However, recently in[10] Cegrell introduced the two new classes: F(Ω) and E(Ω). He has shown that theclass E(Ω) is the natural domain of definition of the complex Monge–Ampere oper-ator and is the biggest class on which the complex Monge–Ampere operator is welldefined and is continuous under decreasing sequences of plurisubharmonic functions([10, Theorem 4.5]). The class E(Ω) contains some important classes of plurisub-harmonic functions, for example, the class of plurisubharmonic functions boundednear boundary introduced and investigated early by Demailly in [17]. By continuingto study the domain of definition of the complex Monge–Ampere operator recentlyBenelkourchi, Guedj, Zeriahi in [6] and [4] introduced the weighted pluricomplexenergy classes Eχ(Ω) and investigated the image of the complex Monge–Ampereoperator acting on this class. One of main results in [6] says that if µ is a positiveBorel measure with µ(Ω) < +∞ which satisfies
µ(K) ≤ Fε(CapΩ(K)),
for all compact subsets K ⊂ Ω and −χ(t) = exp(nH−1(t)/2) then there existsϕ ∈ Eχ(Ω) with (ddcϕ)n = µ (see [6, Theorem 5.1]). After that, by characterizingthe class Eχ(Ω) in terms of the speed of decrease of the capacity of sublevel sets in[4] Belnekourchi has given a description of the range of the complex Monge–Ampereoperator (ddc.)n in the class Eχ(Ω) (see [4, Theorem 5.1]). Recently, in [14] and,more general, in [19] it was proved that if µ is a positive and finite measure on ahyperconvex domain Ω in Cn such that µ(P ) = 0 for all pluripolar sets P ⊂ Ω thenthere exists u ∈ Eχ(Ω) such that −χ(u)(ddcu)n = µ. Continuing the direction ofstudy of the above authors in the paper we investigate the existence of solutionsof the complex Monge–Ampere type equation in the class Eχ(Ω) and the complexMonge–Ampere equation on the class E(Ω). Namely in Theorem 3.2 below we provewithout assumption about vanishing on pluripolar sets that if µ is a positive andfinite with µ ≤ −χ(w)(ddcw)n, w ∈ Eχ(Ω) then there exists u ∈ Eχ(Ω) such that−χ(u)(ddcu)n = µ. Next, as an application of this result we show that if µ is apositive measure on a hyperconvex domain Ω ⊂ Cn such that
∫Ω −ψdµ < +∞
for some ψ ∈ E0(Ω) and for all z ∈ Ω there exists a neighborhood Wz of z andvz ∈ E(Wz) with µ ≤ (ddcvz)n on Wz then there exist a function ϕ ∈ N (Ω)such that (ddcϕ)n = µ. Moreover, by an example we indicate that the condition∫Ω −ψdµ < +∞ is sharp.
The paper is organized as follows. Beside the introduction the paper has threesections. In Sec. 2, we recall some pluricomplex energy classes: E0(Ω),F(Ω),N (Ω),E(Ω) and the class Eχ(Ω) and give more some results concerning with the classEχ(Ω). Section 3 is devoted to study the Monge–Ampere type equation in the classEχ(Ω). We show that if there exist a subsolution of the equation −χ(u)(ddcu)n = µ
then this equation is solvable in the class Eχ(Ω). Section 4 deals with the Dirichletproblem in the class E(Ω). We prove that if this problem has subsolutions locally
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then it is solvable in the class N (Ω). The paper is ended by an example showingthat the condition
∫Ω −ψdµ < +∞, ψ ∈ E0(Ω) in Proposition 4.3 is sharp.
2. Weighted Energy Classes
Some elements of pluripotential theory that will be used throughout the paper canbe found in [1, 7, 8, 15–18, 25]. First, we recall the definition of some pluricomplexenergy classes of plurisubharmonic functions introduced and investigated in [9, 10]and the class Eχ(Ω) introduced in [6].
Through this paper by Ω we denote a hyperconvex domain in Cn. That is anopen bounded connected set Ω ⊂ Cn and there exists a negative plurisubharmonicexhaustion function for Ω. By PSH−(Ω) we denote the set of negative plurisubhar-monic functions on Ω.
2.1. Cegrell’s classes
Following [9, 10] we define the pluricomplex energy classes of PSH−(Ω):
E0 = E0(Ω) =ϕ ∈ PSH−(Ω) ∩ L∞(Ω) : lim
z→∂Ωϕ(z) = 0,
∫Ω
(ddcϕ)n <∞,
F = F(Ω) =ϕ ∈ PSH−(Ω) : ∃ E0 ϕj ϕ, sup
j
∫Ω
(ddcϕj)n <∞,
and
E = E(Ω) =ϕ ∈ PSH−(Ω) : ∀ z0 ∈ Ω, ∃ a neighbourhood ω z0,
E0 ϕj ϕ on ω, supj
∫Ω
(ddcϕj)n <∞.
The following inclusions are clear: E0(Ω) ⊂ F(Ω) ⊂ E(Ω).In [10], Cegrell has proved that E(Ω) is the biggest class on which the complex
Monge–Ampere operator (ddc.)n can be well defined. Moreover, if u ∈ E(Ω) andK Ω then we can find uK ∈ F(Ω) such that u = uK on K.
2.2. The class N (Ω)
We recall the class N (Ω) introduced in [11]. Let Ω be a hyperconvex domain inCn and Ωjj≥1 a fundamental sequence of Ω. This is an increasing sequence ofstrictly pseudoconvex subsets Ωj of Ω such that Ωj Ωj+1 and
⋃∞j=1 Ωj = Ω. Let
ϕ ∈ PSH−(Ω). For each j ≥ 1, put
ϕj = supu : u ∈ PSH(Ω), u ≤ ϕ on Ω\Ωj.As in [11], the function ϕ = (limj→∞ ϕj)∗ ∈ PSH(Ω) and ϕ ∈ MPSH(Ω), where
MPSH(Ω) denotes the set of maximal plurisubharmonic functions on Ω. Set
N = N (Ω) = ϕ ∈ E : ϕ = 0
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or equivalently,
N = N (Ω) = ϕ ∈ PSH−(Ω) : ϕj ↑ 0 a.e.It is easy to see that E0(Ω) ∪ F(Ω) ⊂ N (Ω).
2.3. Functions associated to measures carried on pluripolar sets
For u ∈ E(Ω) we write µu = 1u=−∞(ddcu)n and define S to be the class ofsimple functions f =
∑mj=1 αj1Ej , αj > 0, where Ej are pairwise disjoint and µu-
measurable, 1Ej is the characteristic function of Ej , such that f has a compactsupport and vanishes outside u = −∞. We write T for functions in S where theEj ’s are compact. If 0 ≤ g is a bounded µu-measurable function then we define
ug = infTf≤g
(supuτ : f ≤ τ, τ is a bounded lower semicontinuous function)∗,
where uτ = supϕ ∈ PSH(Ω) : ϕ ≤ τ1/nu. We need the following proposition.
Proposition 2.1. Let u ∈ E(Ω) and let f, g ≥ 0 be bounded µu-measurable func-tions that vanish outside u = −∞. Then the following assertions hold.
(a) if M is a positive constant then uMg = n√Mug.
(b) if f ≤ g then uf ≥ ug.(c) ug ∈ E(Ω) and (ddcug)n = g(ddcu)n.(d) if u ∈ F(Ω) then ug ∈ F(Ω).
Proof. It is easy to see that (a), (b) and (d) follow from the definition of ug. Nowwe prove (c). Let M > 0 such that 0 ≤ Mg ≤ 1. By [1, Theorem 4.8] we haveuMg ∈ E(Ω) and (ddcuMg)n = Mg(ddcu)n. Since uMg = n
√Mug so ug = uMg
n√M∈
E(Ω), and hence,
(ddcug)n =(ddcuMg)n
M=Mg(ddcu)n
M= g(ddcu)n
and the desired conclusion follows.
2.4. Capacities of a Borel subset
Following [24, 27] we define the Cn-capacity and Cn−1-capacity. Let E ⊂ Ω be aBorel subset. Then
Cn(E) = Cn(E,Ω) = sup∫
E
(ddcu)n : u ∈ PSH(Ω),−1 < u < 0
and
Cn−1(E) = Cn−1(E,Ω) = supCn−1(K) : K is a compact subset of E,where for the compact subset K we set
Cn−1(K) = sup∫
K
(ddcu)n−1 ∧ ddc|z|2 : u ∈ PSH(Ω),−1 < u < 0.
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Let q = n or n − 1. Following [27] we say that the sequence uj ⊂ PSH(Ω)converges to u ∈ PSH(Ω) in Cq-capacity as j tends to +∞ if for every compactsubset K of Ω and every ε > 0 we have
limj→+∞
Cq(z ∈ K : |uj(z) − u(z)| > ε) = 0.
2.5. The class Eχ(Ω)
Next we recall the weighted pluricomplex energy class Eχ(Ω) introduced and inves-tigated by Benelkourchi, Guedj and Zeriahi in [6] recently.
Definition 2.1. Let χ : R− → R
− be an increasing function. We denote by Eχ(Ω)the set of all functions u ∈ PSH−(Ω) for which there exists a sequence [uj] ⊂ E0(Ω)decreasing toward u in Ω and satisfying
supj∈N
∫Ω
[−χ(uj)](ddcuj)n < +∞.
Note that if χ(t) < 0 for all t < 0 then by [5, Theorem 2.7] we have
Eχ(Ω) =u ∈ N (Ω) :
∫Ω
[−χ(u)](ddcu)n < +∞.
Moreover, [19, Corollary 3.3] implies that if χ ≡ 0 then Eχ(Ω) ⊂ E(Ω) and,hence, in this case the complex Monge–Ampere operator (ddc.)n is well defined onEχ(Ω).
We need the following result for the class Eχ(Ω).
Proposition 2.2. Let χ : R− → R− be an increasing continuous function,χ(−∞) > −∞. Assume that uj, u ⊂ E(Ω), uj ≥ v, ∀ j ≥ 1 for some v ∈ E(Ω) .Then the following assertions hold.
(a) if uj → u in Cn−1-capacity then lim infj→∞[−χ(uj)](ddcuj)n ≥ −χ(u)(ddcu)n.(b) if uj → u in Cn-capacity then −χ(uj)(ddcuj)n → −χ(u)(ddcu)n weakly.
Proof. (a) Let f ∈ C∞0 (Ω) with 0 ≤ f ≤ 1. We need to prove that
lim infj→∞
∫Ω
−fχ(uj)(ddcuj)n ≥∫
Ω
−fχ(u)(ddcu)n. (2.1)
By [10] we choose a sequence ϕk ⊂ E0 ∩ C(Ω) such that ϕk u and ϕk > u
on Ω. Fix k ∈ N∗, by [22] we can find j0 such that uj ≤ ϕk on supp f for all j ≥ j0.Hence, by using the main result in [12] we get
lim infj→∞
∫Ω
−fχ(uj)(ddcuj)n ≥ lim infj→∞
∫Ω
−fχ(ϕk)(ddcuj)n =∫
Ω
−fχ(ϕk)(ddcu)n,
for all k ≥ 1. Letting k → ∞ and using Lebegues monotone convergence theoremit follows that
lim infj→∞
∫Ω
−fχ(uj)(ddcuj)n ≥∫
Ω
−fχ(u)(ddcu)n.
Thus, lim infj→∞[−χ(uj)](ddcuj)n ≥ −χ(u)(ddcu)n.
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(b) We can assume that χ(−∞) = −1. Let f ∈ C∞0 (Ω) with 0 ≤ f ≤ 1. First
we prove that
lim supj→∞
∫Ω
−fχ(uj)(ddcuj)n ≤∫
Ω
−fχ(u)(ddcu)n. (2.2)
Fix k ∈ N∗. By quasicontinuity of u, v we can take an open subset Gk of Ω and afunction uk ∈ C(Ω) such that Cn(Gk) < 1
2k and uk = u on Ω\Gk and v > −ak onsupp f\Gk with some ak > 0. For every ε > 0, by [26, Theorem 4.1], we have
∫Ω
−fχ(uj)(ddcuj)n =∫
Ω\Gk
−fχ(uj)(ddcuj)n +∫
Gk
−fχ(uj)(ddcuj)n
≤∫
Ω\Gk
−fχ(uj)(ddcuj)n +∫
Gk
f(ddcuj)n
≤∫uj≤u−ε\Gk
−fχ(uj)(ddcuj)n
+∫uj>u−ε\Gk
−fχ(uj)(ddcuj)n +∫
Gk
f(ddcuj)n
≤∫uj≤u−ε\Gk
f(ddcuj)n +∫
Ω\Gk
−fχ(u− ε)(ddcuj)n
+∫
Ω
−fhGk,Ω(ddcuj)n
≤∫uj<u−ε\Gk
(ddc max(uj ,−ak))n
+∫
Ω\Gk
−fχ(uk − ε)(ddcuj)n +∫
Ω
−fhGk,Ω(ddcuj)n
≤ ankCapn(uj < u− ε ∩ supp f)
+∫
Ω\Gk
−fχ(uk − ε)(ddcuj)n +∫
Ω
−fhGk,Ω(ddcuj)n.
Therefore, by [21, Theorem 3.1] and letting j → ∞, we get
lim supj→∞
∫Ω
−fχ(uj)(ddcuj)n
≤∫
Ω\Gk
−fχ(uk − ε)(ddcu)n +∫
Ω
−fhGk,Ω(ddcu)n.
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Letting ε→ 0 we infer that
lim supj→∞
∫Ω
−fχ(uj)(ddcuj)n
≤∫
Ω\Gk
−fχ(uk)(ddcu)n +∫
Ω
−fhGk,Ω(ddcu)n
≤∫
Ω\u=−∞−fχ(u)(ddcu)n +
∫Ω
−fhS∞m=k Gm,Ω(ddcu)n. (2.3)
On the other hand, since⋃∞
m=k Gm G as k → ∞ and
Cn(G) ≤ limk→∞
Cn
( ∞⋃m=k
Gm
)≤ lim
k→∞
∞∑m=k
Cn(Gm) ≤ limk→∞
12k−1
= 0
so we get hS∞m=k Gm,Ω 0 on Ω\P , P is a pluripolar set. Now, letting k → ∞ in
(2.3), we get
lim supj→∞
∫Ω
−fχ(uj)(ddcuj)n
≤∫
Ω\u=−∞−fχ(u)(ddcu)n +
∫P
f(ddcu)n
≤∫
Ω\u=−∞−fχ(u)(ddcu)n +
∫u=−∞
−fχ(u)(ddcu)n
=∫
Ω
−fχ(u)(ddcu)n.
Thus, (2.2) is proved. Moreover, since uj → u in Cn-capacity so uj → u in Cn−1-capacity. Hence, by part (a) we have
lim infj→∞
∫Ω
−fχ(uj)(ddcuj)n ≥∫
Ω
−fχ(u)(ddcu)n.
Combining this with (2.2) we get
limj→∞
∫Ω
−fχ(uj)(ddcuj)n =∫
Ω
−fχ(u)(ddcu)n,
for every f ∈ C∞0 (Ω) with 0 ≤ f ≤ 1. Hence, −χ(uj)(ddcuj)n → −χ(u)(ddcu)n
weakly. The proof is complete.
3. The Monge–Ampere Type Equation in the Class Eχ(Ω)
The section is devoted to study the Monge–Ampere type equation in the classEχ(Ω). As well-known in [14] if Ω is a bounded hyperconvex domain in Cn andχ : R− → R− is an increasing function satisfying χ(−∞) = −∞, χ(0) = 0 and µ
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is a positive and finite measure in Ω such that µ vanishes on pluripolar sets of Ω.Then there exists a function u ∈ Eχ(Ω) such that
−χ(u)(ddcu)n = µ.
After that under the same hypothesis for µ [19, Theorem 4.7] gives an extensionof the above result for the class Eχ(H,Ω) and the Monge–Ampere type equation
−χ(u(z), z)(ddcu)n = µ.
In Theorem 3.1 below without the hypothesis on vanishing of the measure µ onpluripolar sets of Ω we show that the Monge–Ampere type equation
−χ(u)(ddcu)n = µ
is solvable in the class Eχ(Ω). First we need the following result which in the caseχ(t) = −1 for all t < 0 gives the part (b) of [26, Proposition 4.3].
Lemma 3.1. Let χ : R− → R− be an increasing continuous function, χ(−∞) >−∞ and let µ be a positive Radon measure such that µ(P ) = 0 for all pluripo-lar subsets P of Ω. Assume that u, v ∈ E(Ω) such that −χ(u)(ddcu)n ≥ µ and−χ(v)(ddcv)n ≥ µ. Then
−χ(max(u, v))(ddc max(u, v))n ≥ µ.
Proof. Since (u = v − ε\u = v = −∞) ∩ (u = v − δ\u = v = −∞) = ∅,∀ ε = δ we may choose εj 0 such that µ(u = v − εj) = 0 for all j ≥ 1. Since χis an increasing function so by [26, Theorem 4.1] we have
−χ(max(u, v − εj))(ddc max(u, v − εj))n
≥ −χ(u)1u>v−εj(ddcu)n − χ(v − εj)1u<v−εj(dd
cv)n
≥ −χ(u)1u>v−εj(ddcu)n − χ(v)1u<v−εj(dd
cv)n
≥ 1u>v−εjµ+ 1u<v−εjµ = µ ∀ j ≥ 1.
Letting j → ∞ and by Proposition 2.2(b) we get the desired conclusion.
The main result of the section is an extension of main theorem in [1] where theoperator Monge–Ampere (ddc.)n is replaced by −χ(.)(ddc.)n.
Theorem 3.1. Let χ : R− → R− be an increasing continuous function with χ(t) <0 for all t < 0 and µ be a Radon measure such that µ(Ω) < +∞. Assume that thereexists a function w ∈ Eχ(Ω) with µ ≤ −χ(w)(ddcw)n. Then there exists a functionu ∈ Eχ(Ω) such that u ≥ w and −χ(u)(ddcu)n = µ.
Proof. We consider the two cases.
Case 1. Assume that χ(−∞) = −∞. Remark 3.4 in [19] implies that w ∈ Ea(Ω) soµ vanishes on pluripolar sets. Hence, by [19, Theorem 4.10] there exists a functionu ∈ Eχ(Ω) such that −χ(u)(ddcu)n = µ. Moreover, [19, Corollaries 3.3 and 4.9]imply that u ≥ w.
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Case 2. χ(−∞) > −∞. By replacing χ by χ−χ(−∞) we can assume that χ(−∞) =
−1. For each finite Radon measure α on Ω which vanishes on pluripolar sets of Ωsuch that suppα Ω, and v ∈ F(Ω) such that supp(ddcv)n Ω and (ddcv)n iscarried by a pluripolar set we set
u = (sup ϕ : ϕ ∈ B(α, v))∗,where
B(α, v) = ϕ ∈ E(Ω) : α ≤ −χ(ϕ)(ddcϕ)n and ϕ ≤ v .We split the proof in this case into three steps.
Step 1. We prove that u ∈ F(Ω) and
−χ(u)(ddcu)n ≥ α+ (ddcv)n. (3.1)
Indeed, first by [19, Theorem 4.10] there exists a function φ ∈ Eχ(Ω) ⊂ N (Ω) suchthat −χ(φ)(ddcφ)n = α. Since supp(ddcφ)n =supp α
−χ(φ) Ω so∫
Ω(ddcφ)n < +∞.Hence φ ∈ Fa(Ω). Moreover, we have (φ+ v) belongs to B(α, v) and thus, φ+ v ≤u ≤ v. Therefore, we have u ∈ F(Ω). By Lemma 3.1 we have max(ϕ, ψ) ∈ B(α, v),∀ϕ, ψ ∈ B(α, v). We will prove that −χ(u)(ddcu)n ≥ α. Indeed, by using theChoquet lemma we infer that there exists a sequence uj ⊂ B(α, v) such that
u =(
supj∈N∗
uj
)∗.
Set uj = maxu1, . . . , uj ∈ B(α, v). We have uj u a.e. Proposition 2.2 impliesthat −χ(uj)(ddcuj)n → −χ(u)(ddcu)n weakly. Hence −χ(u)(ddcu)n ≥ α and weget u ∈ B(α, v). We have
−χ(u)(ddcu)n = −χ(u)1u>−∞(ddcu)n − χ(u)1u=−∞(ddcu)n
= −χ(u)1u>−∞(ddcu)n + 1u=−∞(ddcu)n.
By [1, Lemma 4.1], we get
−χ(u)(ddcu)n ≥ 1u=−∞(ddcu)n ≥ 1v=−∞(ddcv)n = (ddcv)n.
Combining above inequalities, we get
−χ(u)(ddcu)n ≥ α+ (ddcv)n.
Step 2. We prove that
−χ(u)(ddcu)n = α+ (ddcv)n.
Choose a hyperconvex domain G Ω such that suppα ∪ supp(ddcv)n G Ω.Next let vj ⊂ E0(Ω) be a decreasing sequence that converges pointwise to v in Gas j → +∞ and supp(ddcvj)n ⊂ G. Now since∫
Ω
[α− χ(vj)(ddcvj)n] ≤∫
Ω
[α+ (ddcvj)n] < +∞
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so by [19, Theorem 4.10] there exists an unique function wj ∈ Eχ(Ω) such that
−χ(wj)(ddcwj)n = α− χ(vj)(ddcvj)n.
We have −χ(φ + vj)(ddc(φ + vj))n ≥ −χ(wj)(ddcwj)n ≥ −χ(vj)(ddcvj)n. From[19, Corollary 4.9] we get φ + vj ≤ wj ≤ vj . It follows that wj ∈ B(α, vj) andwj ∈ F(Ω) so if we put
uj = (supϕ : ϕ ∈ B(α, vj))∗ ∈ F(Ω),
then uj ≥ wj for all j ≥ 1 and uj decreases pointwise to u, as j → +∞. Weprove that uj −wj → 0 in Cn−1-capacity. Fix a strictly plurisubharmonic functionh0 ∈ E0 ∩ C∞(Ω). For ε > 0 and j0 ≥ 1, by [26, Proposition 3.4] we have
lim supj→∞
Cn−1(uj − wj > ε)
= lim supj→∞
(sup
∫uj−wj>ε
ddch0 ∧ (ddch)n−1 : h ∈ PSH(Ω),−1 ≤ h ≤ 0
)
≤ lim supj→∞
(sup
1εn
∫uj−wj>ε
(uj − wj)nddch0 ∧ (ddch)n−1 :
h ∈ PSH(Ω),−1 ≤ h ≤ 0
)
≤ lim supj→∞
(sup
1εn
∫Ω
(uj − wj)nddch0 ∧ (ddch)n−1 :
h ∈ PSH(Ω),−1 ≤ h ≤ 0)
≤ lim supj→∞
n!εn
∫Ω
−h0[(ddcwj)n − (ddcuj)n]
≤ lim supj→∞
n!εn
∫Ω
−h0(ddcwj)n +n!εn
∫Ω
h0(ddcu)n
≤ lim supj→∞
n!εn
∫Ω
−h0−χ(wj)(ddcwj)n
−χ(uj)+n!εn
∫Ω
h0(ddcu)n
≤ lim supj→∞
n!εn
∫Ω
−h0α+ (ddcvj)n
−χ(uj)+n!εn
∫Ω
h0(ddcu)n
≤ lim supj→∞
n!εn
∫Ω
−h0α+ (ddcvj)n
−χ(uj0)+n!εn
∫Ω
h0(ddcu)n
≤ n!εn
∫Ω
−h0α+ (ddcv)n
−χ(uj0)+n!εn
∫Ω
h0(ddcu)n,
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where the last inequality follows from suppα∪ supp(ddcvj)n ⊂ G. Letting j0 → ∞by Lebegues monotone convergence theorem and Step 1 we get
lim supj→∞
Cn−1(uj − wj > ε)
≤ lim supj0→∞
n!εn
∫Ω
−h0α+ (ddcv)n
−χ(uj0)+n!εn
∫Ω
h0(ddcu)n
=n!εn
∫Ω
−h0
[α+ (ddcv)n
−χ(u)− (ddcu)n
]≤ 0.
Hence, wj → u in Cn−1-capacity so by Proposition 2.2 we have
lim infj→∞
−χ(wj)(ddcwj)n ≥ −χ(u)(ddcu)n.
This implies that α+ (ddcv)n ≥ −χ(u)(ddcu)n. Combining this with (3.1), we get
−χ(u)(ddcu)n = α+ (ddcv)n.
Step 3. Now we prove the main theorem. By [10, Theorem 5.11] we have a decom-position µ = α + β, where α and β are Radon measures defined on Ω suchthat α vanishes on all pluripolar sets and β is carried by a pluripolar set. Fromβ ≤ −χ(w)(ddcw)n ≤ (ddcw)n and [1, Theorem 4.14] we can find v ∈ N (Ω) suchthat v ≥ w and (ddcv)n = β where (ddcv)n is carried by a pluripolar set v = −∞.
Choose a sequence Ωk, Ωk Ω, Ωk Ω as k ∞. By Proposition 2.1 we canfind a decreasing sequence vk ∈ F(Ω), vk ≥ v and (ddcvk)n = 1Ωk
(ddcv)n = 1Ωkβ.
Set αk = 1Ωkα and
uk = sup ϕ : ϕ ∈ B(αk, vk) .By Step 2, we have −χ(uk)(ddcuk)n = αk + (ddcvk)n. It is easy to see that w ∈B(αk, vk) so uk ≥ w for every k, and hence, uk u ≥ w. Since αk + (ddcvk)n →α+ (ddcv)n weakly so by Proposition 2.2 we get
−χ(u)(ddcu)n = α+ (ddcv)n = µ.
The proof is complete.
Corollary 3.1. Let χ : R− → R− be an increasing continuous function with χ(t) <0 for all t < 0 and let U G ⊂ Ω be bounded hyperconvex domains. Then forevery u ∈ Eχ(G) there exists a function u ∈ Eχ(Ω) such that −χ(u)(ddcu)n =1U [−χ(u)](ddcu)n on Ω.
Proof. We consider the two cases.
Case 1. χ(−∞) = −∞. In this case by [19, Remark 3.4] we have Eχ(Ω) ⊂ Ea(Ω).Hence µ = 1U [−χ(u)](ddcu)n is a non-negative measure vanishing on pluripolarsets of Ω and [19, Theorem 4.10] implies that there exists a function u ∈ Eχ(Ω)such that −χ(u)(ddcu)n = 1U [−χ(u)](ddcu)n.
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Case 2. χ(−∞) > −∞. Then by [19, Corollary 3.3] we have Eχ(G) ⊂ N (G) ⊂ E(G)so there exists u1 ∈ F(G) such that u = u1 on U . We set
u2 = supϕ ∈ PSH−(Ω) : ϕ ≤ u1 on G.Then u2 ∈ F(Ω) and [20, Lemma 4.5] implies that (ddcu2)n ≤ 1G(ddcu1)n on Ω.Moreover, since u2 ≤ u1 in G so by [1, Proposition 4.1] we get
1u2=−∞(ddcu2)n ≥ 1u1=−∞(ddcu1)n on G.
Hence, we have
1u2=−∞(ddcu2)n = 1G1u1=−∞(ddcu1)n on Ω.
On the other hand, the measure 1U∩u>−∞[−χ(u)](ddcu)n vanishes on pluripolarsets of Ω and
∫Ω
1U∩u>−∞[−χ(u)](ddcu)n ≤ ∫G[−χ(u)](ddcu)n < +∞ so by
[19, Theorem 4.10] we can find w ∈ Eχ(Ω) such that
−χ(w)(ddcw)n = 1U∩u>−∞[−χ(u)](ddcu)n.
We claim that w ∈ F(Ω). Indeed, put w = supϕ ∈ PSH−(Ω) : ϕ ≤ w on G. It iseasy to see that w ∈ F(Ω), w ≥ w and w = w in G. Since
−χ(w)(ddcw)n = −1Uχ(w)(ddcw)n = −1Uχ(w)(ddcw)n ≤ −χ(w)(ddcw)n
so by [19, Theorem 4.8], and hence, we get w ≥ w. Hence, w = w ∈ F(Ω).Now, since u1 = u on U so we have
1U [−χ(u)](ddcu)n = 1U∩u>−∞[−χ(u)](ddcu)n + 1U∩u=−∞[−χ(u)](ddcu)n
= 1U∩u>−∞[−χ(u)](ddcu)n + 1U∩u1=−∞[−χ(u1)](ddcu1)n
≤ −χ(w)(ddcw)n − χ(u2)(ddcu2)n
≤ −χ(w + u2)[(ddcw)n + (ddcu2)n]
≤ −χ(w + u2)[(ddc(w + u2)]n.
Moreover, since w, u2 ∈ F(Ω) so w + u2 ∈ F(Ω). Hence, w + u2 ∈ Eχ(Ω) andTheorem 3.1 implies that there exists a u ∈ Eχ(Ω) such that −χ(u)(ddcu)n =1U [−χ(u)](ddcu)n on Ω. The proof is complete.
Corollary 3.2. Let χ : R− → R− be an increasing continuous function withχ(t) < 0 for all t < 0 and χ(−∞) > −∞. Assume that v ∈ F(Ω), andf ∈ L1
loc((ddcv)n) with f ≥ 0. Then there exists a decreasing sequence uj ⊂ F(Ω)
such that supp(ddcuj)n Ω and −χ(uj)(ddcuj)n f(ddcv)n, as j → +∞.
Proof. Choose a sequence Ωj such that Ωj Ω, j ≥ 1 and Ωj Ω as j ∞.For each j ∈ N∗, put gj = 1Ωj∩v=−∞ min(f, j) and uj = sup ϕ : ϕ ∈ B(αj, v
gj )∗,where αj = 1Ωj∩v>−∞ min(f, j)(ddcv)n. By Proposition 2.1, we have vgj ∈ F(Ω).
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Hence, by Step 1 in the proof of Theorem 3.1, we have uj ∈ F(Ω). Therefore, byStep 2 in the proof of Theorem 3.1 we have
−χ(uj)(ddcuj)n = αj + (ddcvgj )n
= 1Ωj∩v>−∞ min(f, j)(ddcv)n
+ 1Ωj∩v=−∞ min(f, j)(ddcv)n
= 1Ωj min(f, j)(ddcv)n. (3.2)
It follows that∫Ω−χ(uj)(ddcuj)n < +∞, and thus, uj ∈ Eχ(Ω). Now, we claim
that uj is a decreasing sequence. Indeed, by Proposition 2.1 we have vgj isdecreasing sequence so uj+1 ≤ vgj+1 ≤ vgj . On the other hand,
−χ(uj+1)(ddcuj+1)n ≥ αj+1 = 1Ωj+1∩v>−∞ min(f, j + 1)(ddcv)n
≥ 1Ωj∩v>−∞ min(f, j)(ddcv)n = αj .
Hence, uj+1 ∈ B(αj, vgj ). Thus, uj+1 ≤ uj and the desired claim is proved. More-
over, by (3.2) we have supp(ddcuj)n Ω and −χ(uj)(ddcuj)n f(ddcv)n, asj +∞. The proof is complete.
4. Applications to the Dirichlet Problem
In the section, we apply the above results to investigating the Dirichlet problem.We show that if the Dirichlet problem is solvable locally then it is solvable globally.First we prove the following.
Proposition 4.1. Assume that µ is a non-negative measure on Ω such that for allz ∈ Ω there exist a neighborhood Wz of z and vz ∈ E(Wz) such that µ ≤ (ddcvz)n
in Wz. Then there exists ϕ ∈ F(Ω) and 0 ≤ f ∈ L1loc((dd
cϕ)n) such that µ =f(ddcϕ)n.
Proof. For each z ∈ Ω, let Uz Gz Wz be hyperconvex domains with z ∈ Uz
and wz ∈ F(Wz) such that wz = vz in Uz. Applying Corollary 3.1 to χ(t) ≡ −1for all t < 0 we can find uz ∈ F(Ω) such that (ddcuz)n = (ddcwz)n = (ddcvz)n ≥ µ
on Uz. Let Ωj be a fundamental increasing sequence of strictly pseudoconvexsubsets of Ω such that Ωj Ωj+1 and
⋃∞j=1 Ωj = Ω. By compactness of Ωj we can
choose ϕj ∈ F(Ω) such that (ddcϕj)n ≥ µ|Ωj. By [13, Lemma 2.5] we can choose a
sequence εj > 0 such that
ϕ =∞∑
j=1
εjϕj ∈ F(Ω).
This implies that µ (ddcϕ)n. Hence, we can find f ∈ L1loc((dd
cϕ)n), f ≥ 0 suchthat µ = f(ddcϕ)n. The proof is complete.
Next, we have the following.
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Proposition 4.2. Let χ : R− → R− be a convex increasing function such thatχ(t) < 0 for all t < 0 and χ(−∞) > −∞. Assume that µ is a non-negative mea-sure on Ω such that µ(Ω) < +∞. Then there exists u ∈ Eχ(Ω) such that −χ(u)(ddcu)n = µ if and only if for all z ∈ Ω there exist a neighborhood Wz of z andvz ∈ E(Wz) such that µ ≤ (ddcvz)n in Wz.
Proof. The necessity is obvious. We give the proof of the sufficiency. By Propo-sition 4.1 and Corollary 3.2 there exists a decreasing sequence uj ⊂ F(Ω) suchthat −χ(uj)(ddcuj)n µ, as j tend to +∞. Put u := limj→+∞ uj. We claim thatu ∈ Eχ(Ω). Indeed, by [10] we choose a sequence vj⊂E0 ∩C(Ω) such that vj u.Put wj = maxvj , uj. Then wj ∈ E0(Ω) and wj u. Moreover, by [1, Lemma 3.3]we have∫
Ω
−χ(wj)(ddcwj)n ≤∫
Ω
−χ(wj)(ddcuj)n ≤∫
Ω
−χ(uj)(ddcuj)n ≤ µ(Ω).
Thus,
supj∈N∗
∫Ω
−χ(wj)(ddcwj)n ≤ µ(Ω) < +∞.
Hence, u ∈ Eχ(Ω) ⊂ E(Ω) and the desired conclusion follows. On the other hand,by Proposition 2.2 we have −χ(uj)(ddcuj)n → −χ(u)(ddcu)n weakly, and hence,−χ(u)(ddcu)n = µ. The proof is complete.
Now we solve the Dirichlet problem in the class E(Ω). Namely we have thefollowing.
Proposition 4.3. Let µ be a non-negative measure on Ω such that∫
Ω−ψdµ < +∞
for some ψ ∈ E0(Ω). Assume that for all z ∈ Ω there exist a neighborhood Wz ofz and vz ∈ E(Wz) such that µ ≤ (ddcvz)n in Wz. Then there exists a functionu ∈ N (Ω) such that (ddcu)n = µ.
Proof. By Proposition 4.1 and Corollary 3.2 there exists a decreasing sequenceuj ⊂ F(Ω) such that (ddcuj)n µ, as j tend to +∞. Put u := limj→+∞ uj. Weclaim that u ∈ E(Ω). Indeed, let U G Ω. Put
vj = supϕ ∈ PSH−(Ω) : ϕ ≤ uj on U ∈ F(Ω).
We have (ddcvj)n = 0 on Ω\U . Moreover, by [1, Lemma 3.3] we have∫Ω
(−ψ)(ddcvj)n ≤∫
Ω
(−ψ)(ddcuj)n ≤∫
Ω
(−ψ)dµ < +∞.
Hence, ∫Ω
(ddcvj)n ≤ 1−supG ψ
∫Ω
(−ψ)dµ < +∞, for every j ≥ 1.
Therefore, by [14, Lemma 2.1], we get
v = limj→∞
vj ∈ F(Ω).
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Since u = v on U so u ∈ E(Ω) and (ddcu) = µ.Next, we claim that u ∈ N (Ω). Indeed, we can assume that −1 ≤ ψ < 0 and ψ
is a strictly plurisubharmonic function. We set
ukj = supϕ ∈ PSH−(Ω) : ϕ ≤ uj on Ω\Ωk,
where Ωk is a fundamental increasing sequence of strictly pseudoconvex subsetsof Ω such that Ωk Ωk+1 and
⋃∞k=1 Ωk = Ω. Since uk
j uk as j → ∞ and uk u
as k → ∞ we can choose a sequence jk → ∞ such that ukjk
converges u a.e. By[26, Proposition 3.4], we get∫
Ω
(−ukjk
)n(ddcψ)n ≤ n!∫
Ω
−ψ(ddcukjk
)n = n!∫
Ω
−ψk−1(ddcukjk
)n,
where
ψk = supϕ ∈ PSH−(Ω) : ϕ ≤ ψ on Ω\Ωk.On the other hand, since uk
jk≥ ujk
so by [1, Lemma 3.3] we get∫Ω
(−ukjk
)n(ddcψ)n ≤ n!∫
Ω
−ψk−1(ddcujk)n ≤ n!
∫Ω
−ψk−1(ddcu)n.
Letting k → ∞ by Lebesgue’s convergence theorem, we get∫Ω
(−u)n(ddcψ)n = 0.
Hence u = 0 and the desired conclusion follows.
The following example shows that the condition∫
Ω−ψdµ < +∞ for some
ψ ∈ E0(Ω) in Proposition 4.3 is sharp.
Example 4.1. We construct a non-negative measure µ on the polydisk ∆n (n ≥ 2),which vanishes on all pluripolar sets such that
∫∆n(−max(ln |z1|, . . . , ln |zn|))pdµ <
+∞ for all p > 1 but there is no function u ∈ E(∆n) satisfying (ddcu)n = µ. We set
ψ = max(ln |z1|, . . . , ln |zn|),uj = max(2j ln |z1|, j ln |z2|, . . . , j ln |zn|,−1),
µ =∞∑
j=1
(ddcuj)n.
We have E0(∆n) uj −1 as j → ∞. We have∫∆n
(−ψ)pdµ
=∞∑
j=1
∫∆n
(−ψ)pddc max(2j ln |z1|,−1) ∧ · · · ∧ ddc max(j ln |zn|,−1)
=∞∑
j=1
∫∆n
(−ψ)p2jjn−1ddc max(
ln |z1|,− 12j
)∧ · · · ∧ ddc
(ln |zn|,−1
j
)
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=∞∑
j=1
∫∆n
jn−1
2j(p−1)ddc max
(ln |z1|,− 1
2j
)∧ · · · ∧ ddc
(ln |zn|,−1
j
)
= (2π)n∞∑
j=1
jn−1
2j(p−1)< +∞,
for all p > 1. Assume that µ = (ddcu)n for some u ∈ E(∆n). By the comparisonprinciple we have uj ≥ u for all j ≥ 1. Letting j → ∞ we get −1 ≥ u. Henceu+ 1 ∈ E(∆n). Replace u by u+ 1 and using above argument we have −1 ≥ u+ 1.Hence −2 ≥ u. By induction we obtain u ≡ −∞. This is impossible.
Acknowledgments
The research was done while the second author is supported by the ANR projectMNGNK, decision No. ANR-10-BLAN-0118. He would like to thank ProfessorAndrei Teleman and the members of the LATP, CMI, Universite de Provence,Marseille, France for their kind hospitality. The authors were also supported by theNAFOSTED program.
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