Subextension of plurisubharmonic functions without changing the Monge Amp ere measures and applications

The aim of the paper is to investigate subextension with boundary values of certain plurisubharmonic functions without changing the MongeAmp`ere measures. From the obtained results we show that if given sequence is convergent in Cn−1capacity then sequence of the MongeAmp`ere measures of subextension is weaklyconvergent. As an application we investigate the Dirichlet problem for a nonnegative measure µ in the class F(Ω, g) without the assumption that µ vanishes on all pluripolar sets

changing the Monge-Amp`ere measures and applications

Le Mau Hai and Nguyen Xuan Hong Department of Mathematics, Hanoi National University of Education,

Hanoi, Vietnam E-mail: mauhai@fpt.vn and xuanhongdhsp@yahoo.com

Abstract The aim of the paper is to investigate subextension with boundary values

of certain plurisubharmonic functions without changing the Monge-Amp` ere measures From the obtained results we show that if given sequence is con-vergent in Cn−1-capacity then sequence of the Monge-Amp` ere measures of subextension is weakly*-convergent As an application we investigate the Dirichlet problem for a non-negative measure µ in the class F (Ω, g) without the assumption that µ vanishes on all pluripolar sets.

Let Ω ⊂ eΩ be domains in Cn and u a plurisubharmonic function on Ω (briefly, u ∈ PSH(Ω)) A function u ∈ PSH(ee Ω) is said to be subextension

of u if for all z ∈ Ω,eu(z) ≤ u(z) The subextension problem in the class

F (Ω) introduced and investigated by Cegrell (see [Ce1]) has recently been studied by Cegrell and Zeriahi In [CeZe] the authors proved that if Ω, eΩ are bounded hyperconvex domains in Cn with Ω b eΩ and u ∈ F (Ω), then there exists u ∈ F (ee Ω) such that eu ≤ u on Ω and

Z

e Ω

(ddcu)e n≤

Z

Ω

(ddcu)n

For the class Ep(Ω), p > 0 the subextension problem was investigated by P.H.Hiep He proved in [H2] that if Ω ⊂ eΩ ⊂ Cn are hyperconvex domains and u ∈ Ep(Ω), p > 0, then there exists a functioneu ∈ Ep(eΩ) such thateu ≤ u

on Ω and ep(eu) =R

e Ω

(−eu)p(ddc

e u)n≤R

Ω

(−u)p(ddcu)n The subextension prob-lem concerning to boundary values was considered in recent years Namely,

2000 Mathematics Subject Classification: 32U05, 32U15, 32W20.

Key words and phrases: complex Monge-Amp` ere operator, plurisubharmonic function, subextension.

1

in 2008, in [CzHe] the authors showed that if Ω1 and Ω2 are two bounded hy-perconvex domains such that Ω1 ⊂ Ω2 ⊂ Cn, n ≥ 1 and u ∈ F (Ω1, F ) with

F ∈ E (Ω1) has subextension v ∈ F (Ω2, G) with G ∈ E (Ω2) ∩ MPSH(Ω2), and

Z

Ω 2 (ddcv)n≤

Z

Ω 1 (ddcu)n,

under the assumption that F ≥ G on Ω1, where F (Ω) (resp E (Ω)) are the classes of unbounded plurisubharmonic functions defined in Section 2 and MPSH(Ω) denotes the set of maximal plurisubharmonic functions on Ω It should be remarked that in results of the above mentioned authors they only achieve estimates on total mass of subextension and of the given function

In the paper not only we try to establish the existence of subextension with given boundary values in the class F (Ω, f ) but also we prove that the Monge-Amp`ere measures of subextension and of given function are not changing Namely, we prove the following:

Theorem 3.4 Let Ω ⊂ eΩ be bounded hyperconvex domains in Cn and let

f ∈ E (Ω) and g ∈ E (eΩ) ∩ M P SH(eΩ) with f ≥ g on Ω Then for every

u ∈ F (Ω, f ) withR

Ω

(ddcu)n < +∞ there existseu ∈ F (eΩ, g) such that eu ≤ u

on Ω and (ddc

e u)n= 1Ω(ddcu)n on eΩ

Note that from our results it is easy to obtain estimates on total mass

in [CeZe] and [CzHe] Next from the above result we give a result on the weak*- convergence of sequence of the Monge-Amp`ere measures of subextension if we assume that the given sequence is convergent in Cn−1 -capacity (see Corolary 3.5 below) Also using the obtained result we in-vestigate the Dirichlet problem for a non-negative measure µ in the class

F (Ω, g), g ∈ E(Ω) ∩ M P SH(Ω) It should be noticed that the above prob-lem was considered early in [Ah] In [Ah] the author solved the Dirichlet problem in the class F (Ω, g) for a non-negative measure µ under assumption that µ vanishes on pluripolar sets In our note we omit this assumption The paper is organized as follows Beside the introduction the paper has three sections In Section 2 we give some elements of pluripotential theory which is necessary for contents presented in the paper We recall some classes

of unbounded plurisubharmonic functions introduced and investigated by Cegrell recently Note that by results of Cegrell in [Ce1] and [Ce2] the Monge-Amp`ere operator is well defined on these classes as a non-negative Radon measure Next, in Section 3 we prove the main results of the paper

We give new proofs about subextension of plurisubharmonic functions with boundary values and show the equality between the Monge-Amp`ere mea-sures of subextension and of given function Section 4 is devoted to apply the above result to investigating the Dirichlet problem

2 Preliminaries

First some elements of pluripotential theory that will be used throughout the paper can be found in [ACCH], [BT], [Kl], [Ko1], [Ko2] and [Xi] Now

we recall some Cegrell’s classes of plurisubharmonic functions (see [Ce1] and [Ce2]) and classes of plurisubharmonic functions with generalized boundary values concerning to Cegrell’s classes Let Ω be an open set in Cn By

P SH−(Ω) we denote the set of negative plurisubharmonic functions on Ω 2.1 Now we assume that Ω is a bounded hyperconvex domain in Cn This means that Ω is a bounded domain in Cn and there exists a plurisub-harmonic function ϕ : Ω −→ (−∞, 0) such that for every c < 0 the set

Ωc = {z ∈ Ω : ϕ(z) < c} b Ω As in [Ce1] we define the following subclasses

of P SH−(Ω):

E0 = E0(Ω) = {ϕ ∈ P SH−(Ω) ∩ L∞(Ω) : lim

z→∂Ωϕ(z) = 0,

Z

Ω

(ddcϕ)n < ∞}

F = F (Ω) =ϕ ∈ P SH−

(Ω) : ∃ E0 3 ϕj & ϕ, sup

j

Z

Ω

(ddcϕj)n< ∞ ,

and

E = E(Ω) =ϕ ∈ P SH−

(Ω) : ∀z0 ∈ Ω, ∃ a neighbourhood ω 3 z0,

E0 3 ϕj & ϕ on ω, sup

j

Z

Ω

(ddcϕj)n< ∞

The following inclusions are obvious: E0 ⊂ F ⊂ E

Next, we recall classes of plurisubharmonic functions with generalized bound-ary values in the class E Let K ∈ {E0, F } Then we say that a plurisubhar-monic function u defined on Ω is in K(Ω, G), G ∈ E if there exists a function

ϕ ∈ K such that

ϕ + G ≤ u ≤ G,

on Ω For systematic and detail study of classes of plurisubharmonic func-tions with generalized boundary values in other classes we refer readers to the paper of [˚AhCz2] Note that functions in K(Ω, G) not necessarily have finite total Monge-Amp`ere mass (see [˚AhCz1])

2.2 Because in this note we also need the class of maximal plurisubharmonic functions so we recall the following definition given in [Bl1]

Definition 2.1 A plurisubharmonic function u on Ω is said to be maximal plurisubharmonic (briefly, u ∈ M P SH(Ω)) if for every v ∈ P SH(Ω), v ≤ u outside a compact subset of Ω implies v ≤ u on Ω

As well known, in [Kl], for the class P SH(Ω) ∩ L∞loc(Ω) of locally bounded plurisubharmonic functions, plurisubharmonic functions in this class are maximal if and only if they satisfy the homogeneous Monge-Amp`ere equa-tion (ddcu)n = 0 In [Bl2] Blocki extended the above result for the class E(Ω)

2.3.We now recall the notion of Cn-capacity of a Borel set and extensions

of this notion, as well as, the convergence in capacity of a sequence of plurisubharmonic functions For related definitions we refer readers to the papers of [Xi] and [Ce3] recently Let Ω be an open set in Cn and E ⊂ Ω a Borel subset Following [BT] we define the Cn-capacity of E as follows:

Cn(E) = Cn(E, Ω) = supn

Z

E

(ddcu)n: u ∈ P SH(Ω), −1 < u < 0o

Extending this notion, in [Xi] Xing introduced the notion of Cn−1-capacity Let E ⊂ Ω be a Borel subset The Cn−1-capacity of E is defined by

Cn−1(E) = Cn−1(E, Ω) = sup

nZ

E

(ddcu)n−1∧ ddc|z|2 :

u ∈ P SH(Ω), −1 < u < 0

o

In [Xi] it is remarked that there exits a constant AΩ > 0 such that Cn−1(E) ≤

AΩCn(E) for all Borel subsets E ⊂ Ω

Next, we deal with the convergence of a sequence of plurisubharmonic func-tions in capacity and recall an important result due to Cegrell in [Ce3] which we need in our proofs late Let {uj, u} be plurisubharmonic functions

in an open set Ω of Cn We say that {uj} converges to u in Cs-capacity,

s = n, n − 1 if for every compact subset K of Ω and every δ > 0 it holds that

lim

j→∞Cs {z ∈ K : |uj(z) − u(z)| > δ}
= 0

By the inequality Cn−1(E) ≤ AΩCn(E) for all E ⊂ Ω it follows that if {uj} converges to u in Cn-capacity then so does in Cn−1-capacity Moreover, by using the quasi-continuity of plurisubharmonic functions in [BT] it is not difficult to prove that if a sequence of plurisubharmonic functions {uj} is increasing (or decreasing) and converges to a plurisubharmonic function u then it converges to u in Cn-capacity An important result proved recently

by Cegrell in [Ce3] is as follows Assume that u0 ∈ E and {uj} ⊂ E is a sequence with u0 ≤ uj for j ≥ 1 If {uj} converges to a plurisubharmonic function u ∈ E in Cn−1-capacity then the sequence of measures {(ddcuj)n} converges to (ddcu)n in the weak*-topology as j → ∞ We shall use this result some times in our proofs in the next section

3 Subextension with boundary values without

The aim of this section is to investigate subextension of plurisubharmonic functions in the class F (Ω, f ), f ∈ E without changing the Monge-Amp`ere measures Before to arrive the main result of the note (Theorem 3.4 below)

we need some auxiliary results

The following proposition is a main tool in the proof of Theorem 3.4 and in section 4

Proposition 3.1 Let Ω ⊂ eΩ be bounded hyperconvex domains in Cn and let g ∈ E (eΩ) ∩ M P SH(eΩ) Assume that u ∈ E (Ω) satisfies the conditions: a) R

Ω

(ddcu)n < +∞

b) (ddcv)n ≤ 1Ω(ddcu)n on eΩ with some v ∈ F (eΩ, g) and v ≤ u on Ω Then there exists eu ∈ F (eΩ, g) such that u ≤ v on ee Ω and (ddc

e u)n =

1Ω(ddcu)n on eΩ

Proof Since (ddcv)n ≤ 1Ω(ddcu)n on eΩ so

1{v=−∞}(ddcv)n ≤ 1Ω∩{u=−∞}(ddcu)n,

on eΩ Moreover, since v ≤ u on Ω so Lemma 4.1 in [ACCH] implies that

1Ω∩{v=−∞}(ddcv)n≥ 1Ω∩{u=−∞}(ddcu)n

on Ω Therefore 1{v=−∞}(ddcv)n= 1Ω∩{u=−∞}(ddcu)non eΩ Put µ = 1Ω(ddcu)n− (ddcv)n on eΩ We notice that µ is a nonnegative measure vanishing on pluripolar sets of eΩ We split the proof into three steps

Step 1 We prove that there exists w ∈ F (eΩ, g) such that w ≤ v and

(ddcw)n ≥ µ + 1{v>−∞}(ddcv)n Indeed, since µ vanishes on all pluripolar sets of eΩ and

µ(eΩ) ≤

Z

Ω

(ddcu)n < +∞

so Lemma 5.14 in [Ce1] implies that there exists v0 ∈ F (eΩ) such that (ddcv0)n= µ Put w = v + v0 We have w ∈ F (eΩ, g), w ≤ v and

(ddcw)n ≥ (ddcv0)n+ (ddcv)n≥ µ + 1{v>−∞}(ddcv)n

Step 2 Put

e

u = (sup{ϕ ∈ E (eΩ) : ϕ ≤ v and (ddcϕ)n≥ µ + 1{v>−∞}(ddcv)n})∗

It is clear that w ≤u ≤ v Hence,e u ∈ F (ee Ω, g) We prove that

(ddcu)n ≥ 1Ω(ddcu)n on eΩ

Indeed, by using the Choquet lemma we infer that there exists a sequence {ϕj} ⊂ E(eΩ) such that ϕj ≤ v, (ddcϕj)n≥ µ + 1{v>−∞}(ddcv)n and

e

u =

 sup

j∈N ∗

ϕj

∗

By Proposition 4.3 in [KH] we can replace ϕj by max{w, ϕ1, , ϕj} Hence,

we can assume that w ≤ ϕj and ϕj % u a.e Therefore, ϕe j → u in Ce n -capacity and by [Ce3] we have (ddcϕj)n→ (ddc

e u)n weakly Thus, (ddceu)n≥ µ + 1{v>−∞}(ddcv)n

The above inequality is equivalent to the following

1{eu>−∞}(ddceu)n≥ µ + 1{v>−∞}(ddcv)n (3.1) Moreover, since eu ≤ v on eΩ so Lemma 4.1 in [ACCH] implies that

1{eu=−∞}(ddceu)n≥ 1{v=−∞}(ddcv)n (3.2) Combining (3.1) and (3.2) we infer that

(ddceu)n≥ µ + (ddcv)n= 1Ω(ddcu)n Step 3 For each j = 1, 2, , put

e

uj = (sup{ϕ ∈ E (eΩ) : ϕ ≤ max(v, g − j), (ddcϕ)n≥ µ + 1{v>g−j}(ddcv)n})∗

It is easy to see that eu ≤uej+1 ≤uej for every j and eu = lim

j→∞euj Now, since

µ + (ddcmax(v, g − j))n vanishes on all pluripolar sets so by Theorem 3.9

in [Ce2] there exists a function wj ∈ F (eΩ, g) such that

(ddcwj)n= µ + (ddcmax(v, g − j))n Since (ddcwj)n≥ (ddcmax(v, g − j))n so Theorem 3.8 in [Ce2] implies that

wj ≤ max(v, g − j) Moreover, by Theorem 4.1 in [KH] we have

(ddcwj)n≥ µ + 1{v>g−j}(ddcmax(v, g − j))n = µ + 1{v>g−j}(ddcv)n Hence, wj ≤euj From Proposition 2.2 in [CzHe] it follows that

Z

Ω

(ddcu)n≤

Z

e Ω

(ddcu)e n = lim

j→∞

Z

e Ω

(ddceuj)n

≤ lim sup

j→∞

Z

e Ω

(ddcwj)n

≤ lim sup

j→∞

Z

e Ω

[µ + (ddcmax(v, g − j))n] =

Z

Ω

(ddcu)n Hence,

(ddceu)n= 1Ω(ddcu)n

The next lemma is important for the proof of Theorem 3.4.

Lemma 3.2 Let Ω ⊂ eΩ be bounded hyperconvex domains in Cn Assume that f ∈ E (Ω) and g ∈ E (eΩ) ∩ M P SH(eΩ) such that f ≥ g + δ on Ω with some δ > 0 Then for every u ∈ E0(Ω, f ) such that

w := (sup{ϕ ∈ P SH−(eΩ) : ϕ ≤ g on eΩ\Ω and ϕ ≤ min(u, g) on Ω})∗ ∈ E(eΩ), the following inequality holds: (ddcw)n ≤ 1Ω(ddcu)n on eΩ

Proof We split the proof into two steps

Step 1 We prove that (ddcw)n = 0 on eΩ\Ω Indeed, since u ∈ E0(Ω, f ) so there exists ψ ∈ E0(Ω) such that ψ +f ≤ u ≤ f on Ω Put U :={ψ < −δ} b

Ω Then ψ ≥ −δ on Ω\U so it is easy to see that min(u, g) = g on Ω\U Since eΩ\U is an open set and g ∈ M P SH(eΩ\U ) so w ∈ M P SH(eΩ\U ) Indeed, let v ∈ P SH−(eΩ \ U ) and v ≤ w outside K b eΩ \ U Put

v1 =

( max(v, w) on eΩ\U

Then v1 ∈ P SH−(eΩ) and v1 ≤ w ≤ g outside K in eΩ By the maximality

of g it follows that v1 ≤ g on eΩ It is easy to see that v1 ≤ min(u, g) on

Ω Hence, by definition of w it follows that v1 ≤ w on eΩ and the desired conclusion follows Thus (ddcw)n= 0 on eΩ\Ω

Step 2 We prove (ddcw)n≤ (ddcu)non Ω First, we prove that (ddcw)n= 0

on {w < min(u, g)} ∩ Ω It is easy to see that

{w < min(u, g)} ∩ Ω = [

a∈Q − ({w < a < min(u, g)} ∩ Ω)

a,b∈Q −

[

ε>0

(({w < a < u − ε < b < g} ∩ Ω) ∪ ({w < a < g − ε < b < u}) ∩ Ω)

Hence, it suffices to prove that (ddcw)n = 0 on {w < a < u − ε < b < g} ∩ Ω The proof is similar for the set {w < a < g − ε < b < u} Let {uj} ⊂ E0(Ω) ∩ C(Ω) with uj & u on Ω and {gj} ⊂ E0(eΩ) ∩ C(eΩ) such that

gj & g on eΩ Put

wj := (sup{ϕ ∈ P SH−(eΩ) : ϕ ≤ gj on eΩ\Ω and ϕ ≤ min(uj, gj) on Ω})∗ ∈ E(eΩ)

We have wj & w as j % ∞ and {w < a} =

∞

S

k=1

{wk < a} Hence, it suffices

to show that (ddcw)n = 0 on {wk < a < u − ε < b < g} ∩ Ω By Corollary 9.2 in [BT] it is easy to see that

(ddcwj)n= 0 on {wj < min(uj, gj)} ∩ Ω

Moreover, {wk < a < u − ε < b < g} ∩ Ω ⊂ {wj < min(uj, gj)} ∩ Ω for every

j ≥ k Hence,

(ddcwj)n= 0 on {wk < a < u − ε < b < g} ∩ Ω,

for every j ≥ k Therefore,

max(g − b, 0)(ddcwj)n= 0 on {wk< a < u − ε < b} ∩ Ω,

for every j ≥ k Hence,

max(u − ε − a, 0) max(g − b, 0)(ddcwj)n= 0 on {wk < a} ∩ {u < b + ε} ∩ Ω,

for every j ≥ k This is equivalent to

(max(u − ε − a, 0) + max(g − b, 0))2− max(u − ε − a, 0)2− max(g − b, 0)2

c

wj)n = 0,

on {wk< a} ∩ {u < b + ε} ∩ Ω for every j ≥ k Therefore, by Corollary 3.3

in [Ce3] we get

(max(u − ε − a, 0) + max(g − b, 0))2− max(u − ε − a, 0)2− max(g − b, 0)2

cw)n = 0,

on {wk< a} ∩ {u < b + ε} ∩ Ω Thus Lemma 4.2 in [KH] implies that

(ddcw)n = 0 on {w < a < u − ε < b < g} ∩ Ω

Now we prove that (ddcw)n ≤ (ddcu)n on {w = min(u, g)} ∩ Ω Indeed,

since {w = min(u, g)} ⊂ {w = g} ∪ {w = u} so it suffices to prove that

(ddcw)n ≤ (ddcu)n on {w = g} ∪ {w = u} Let K be a compact set in

{w = g} Since K b {w +1

j > g} for every j so by Theorem 4.1 in [KH] we have

Z

K

(ddcw)n= lim

j→∞

Z

K

(ddcmax(w + 1

j, g))

n

≤ Z

K

(ddcmax(w, g))n =

Z

K

(ddcg)n

Hence, we have (ddcw)n ≤ (ddcg)n = 0 ≤ (ddcu)n on {w = g} Similarly,

we also (ddcw)n≤ (ddcu)n on {w = u} Therefore, (ddcw)n ≤ (ddcu)non Ω

The proof is complete

Remark 3.3 From the proof of the above lemma we have the following

Assume that Ω is a bounded hyperconvex domain in Cn and µ is a

non-negative measure in Ω Assume that u, v ∈ E (Ω) such that (ddcu)n ≤ µ,

(ddcv)n≤ µ and

w := (sup{ϕ ∈ P SH−(Ω) : ϕ ≤ min(u, v) on Ω})∗ ∈ E(Ω)

Then (ddcw)n≤ µ

Now we are position to state the main result of the paper Note that in our

result, in contract to the result in [CeZe], the assumption on Ω b eΩ is not

necessary At the same time, compared with Theorem 1.1 in [CzHe], we

ob-tain a better relation between the Monge-Amp`ere measures of subextension

and a given function

Theorem 3.4 Let Ω ⊂ eΩ be bounded hyperconvex domains in Cn and let

f ∈ E (Ω) and g ∈ E (eΩ) ∩ M P SH(eΩ) with f ≥ g on Ω Then for every

u ∈ F (Ω, f ) with R

Ω

(ddcu)n< +∞ there exists eu ∈ F (eΩ, g) such that eu ≤ u

on Ω and (ddc

e u)n= 1Ω(ddcu)n on eΩ

Proof By Proposition 3.1 it suffices to construct a function v ∈ F (eΩ, g) such that v ≤ u on Ω and (ddcv)n≤ 1Ω(ddcu)n on eΩ

First, we prove that there exists w ∈ F (eΩ, g) such that w ≤ u on Ω Indeed, since u ∈ F (Ω, f ) so there exists u0 ∈ F (Ω) such that u0+ f ≤ u ≤ f on

Ω By Lemma 4.5 in [H2] there exist eu0 ∈ F (eΩ) such that eu0 ≤ u0 on Ω Put w := ue0 + g We have w ∈ F (eΩ, g) Moreover, since f ≥ g on Ω so

w ≤ u0+ f ≤ u on Ω

Now, since u ∈ F (Ω, f ) so there exists {uj} ⊂ E0(Ω, f ) such that uj & u as

j % ∞ Choose a sequence δj & 0 Put gj = g − δj and

vj,k := (sup{ϕ ∈ P SH−(eΩ) : ϕ ≤ gk on eΩ\Ω and ϕ ≤ min(uj, gk) on Ω })∗

It is easy to see that w − δk ≤ vj,k on eΩ so vj,k ∈ E(eΩ) Moreover, {vj,k}k≥1

is increasing as k % ∞ Lemma 3.2 implies that (ddcvj,k)n≤ 1Ω(ddcuj)n on e

Ω Put vj = ( lim

k→∞vj,k)∗ We have vj,k % vj a.e on eΩ then by [Ce3] we get (ddcvj)n≤ 1Ω(ddcuj)n on eΩ

On the other hand, it is easy to see that vj & v ∈ E(eΩ) so again by [Ce3]

we get (ddcvj)n → (ddcv)n weakly in eΩ Moreover, by Lemma 3.1 in [Ce1] and Corollary 3.4 in [ACCH] it follows that 1Ω(ddcuj)n → 1Ω(ddcu)n weakly

in eΩ Therefore,

(ddcv)n≤ 1Ω(ddcu)n on eΩ

Finally, since w ≤ v ≤ g on eΩ so v ∈ F (eΩ, g) and v ≤ u on Ω and the desired conclusion follows

From the above theorem we have the following corollary which deals with the weak*- convergence of sequence of the Monge-Amp`ere measures of subex-tension when the given sequence is convergent in Cn−1-capacity

Corollary 3.5 Let Ω ⊂ eΩ be bounded hyperconvex domains in Cn and

f ∈ E (Ω), g ∈ E (eΩ) ∩ M P SH(eΩ) such that f ≥ g on Ω Then for ev-ery sequence {uj, u0} ⊂ F (Ω, f ) such that uj ≥ u0 for all j ≥ 1 and R

Ω

(ddcuj)n < +∞,R

Ω

(ddcu0)n < +∞, uj is convergent to u0 in Cn−1-capacity

on Ω, sequences of subextension {euj,ue0} ⊂ F (eΩ, g) of {uj, u0} which ex-ist as in Theorem 3.4, satisfy (ddc

e

uj)n is weakly*-convergent to (ddc

e

u0)n as

j → ∞

Proof First we show that for every ϕ ∈ P SH−(Ω) ∩ L∞(Ω) then

lim

j→∞

Z

Ω

ϕ(ddcuj)n=

Z

Ω

ϕ(ddcu0)n (3.3)

Indeed, from the hypothesis it follows that uj is convergent to u0 on Ω according to the Lebesgue measure dVn on Cn Hence, there exists a subse-quence of {ujk}k≥1 of the sequence {uj}j≥1 such that ujk is convergent to u0 a.e on Ω Without loss of generality we may assume that uj is convergent

to u0 a.e on Ω Put vj := (sup{us : s ≥ j})∗ We have uj ≤ vj and, hence,

vj ∈ F (Ω, f ), vj & u0 as j → ∞ Corollary 3.4 in [ACCH] implies that if

ϕ ∈ P SH−(Ω) ∩ L∞(Ω) then

lim

j→∞

Z

Ω

ϕ(ddcvj)n =

Z

Ω

ϕ(ddcu0)n Moreover, since u0 ≤ uj ≤ vj on Ω so by Lemma 3.3 in [ACCH] we have

Z

Ω

ϕ(ddcvj)n ≥

Z

Ω

ϕ(ddcuj)n≥

Z

Ω

ϕ(ddcu0)n Hence, we get

lim

j→∞

Z

Ω

ϕ(ddcvj)n = lim

j→∞

Z

Ω

ϕ(ddcuj)n=

Z

Ω

ϕ(ddcu0)n,

and (3.3) is proved

Now, by Theorem 3.4 there exists sequence of subextension {uej,eu0} ⊂

F (eΩ, g) of the sequence {uj, u0} such that euj ≤ uj, eu0 ≤ u0 on Ω and (ddc

e

uj)n = 1Ω(ddcuj)n, (ddc

e

u0)n = 1Ω(ddcu0)n We prove that (ddc

e

uj)n is weakly*-convergent to (ddc

e

u0)n Indeed, assume that χ ∈ C0∞(eΩ) By Lemma 3.1 in [Ce1] there exists ϕ1, ϕ2 ∈ E0(eΩ) such that χ = ϕ1− ϕ2 We have

lim

j→∞

Z

e Ω

χ(ddceuj)n = lim

j→∞

Z

Ω

ϕ1(ddcuj)n− lim

j→∞

Z

Ω

ϕ2(ddcuj)n

= Z

Ω

ϕ1(ddcu0)n−

Z

Ω

ϕ2(ddcu0)n

= Z

e Ω

χ(ddceu0)n, and we get the required conclusion

In this section we give an application of Theorem 3.4 to solving the Dirichlet problem in the class F (Ω, g), g ∈ M P SH(Ω) ∩ E (Ω) without assumption that the measure µ vanishes on all pluripolar sets

Proposition 4.1 Let Ω be a bounded hyperconvex domain in Cn and let

µ be a non-negative measure in Ω with µ(Ω) < +∞ Assume that g ∈ E(Ω) ∩ M P SH(Ω) Then there exists u ∈ F (Ω, g) with (ddcu)n = µ in Ω

if and only if for every hyperconvex domain U b Ω there exists uU ∈ E(U ) such that (ddcuU)n = µ on U